The Quadratic Reciprocity Law: A Collection of Classical Proofs

The Quadratic Reciprocity Law

Oswald Baumgart

The Quadratic Reciprocity Law A Collection of Classical Proofs Edited and Translated by Franz Lemmermeyer

Oswald Baumgart Translated by Franz Lemmermeyer Jagstzell, Germany

ISBN 978-3-319-16282-9 DOI 10.1007/978-3-319-16283-6

ISBN 978-3-319-16283-6 (eBook)

Library of Congress Control Number: 2015938466 Mathematics Subject Classification (2010): 11A15, 11T22, 11T24 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG and Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com)

Translator’s Preface

This book contains a translation of Baumgart’s thesis Über das quadratische Reciprocitätsgesetz. Eine vergleichende Darstellung der Beweise (On the Quadratic Reciprocity Law. A Comparative Presentation of its Proofs) written in Göttingen a hundred years after Legendre’s publication of the reciprocity law. Its aim was to present and compare the proofs of the quadratic reciprocity law known to him in 1885. The introduction to this book, which was to give a detailed history of the number theory leading up to the statement and the first proofs of the quadratic reciprocity law, eventually evolved into a book of its own and will be published separately. Baumgart’s thesis consists of two parts. The first part gives a very brief history of the development up to Legendre and then presents some different proofs of the quadratic reciprocity law; Baumgart distinguished Gauss’s first proof by induction, proofs by Gauss’s Lemma, by complex analysis, by cyclotomy, and by quadratic forms. The last two chapters deal with supplementary laws and algorithms for computing Legendre symbols. In the second part, Baumgart compares the principles of these proofs. Although Baumgart’s thesis does not fill the gap left by the loss of Cooper’s contribution to Dickson’s history, the planned fourth volume1 on the quadratic reciprocity law, it may perhaps serve as a stepping stone until such a history of quadratic residues is written. A closer look at the proofs of the quadratic reciprocity law discussed by Baumgart reveals that only a few of them compare favorably – in terms of simplicity and elegance – with some of the proofs found in recent years. But this only emphasizes that we should continue striving for beauty even in time-honored subjects such as quadratic reciprocity. In addition I would like to call the readers’ attention to the fact that only a few proofs so far have been transferred to the case of the (quadratic) reciprocity law in polynomial rings over finite fields.

1

See [22]. v

vi

Translator’s Preface

Comments on the Translation In general, I have corrected obvious mistakes without comment. In addition, I have filled a large gap in the presentation of one of Kummer’s proofs. I have also collected the references in the bibliography at the end and have added references to collected works in order to give readers easier access to the original articles. Finally I added comments to the chapters of Part II (“Notes”) in which I have briefly sketched the later developments. In an appendix I have supplied detailed references to all the proofs of the quadratic reciprocity law in the integers that I was aware of at the time of writing. Jagstzell, Germany December 2014

Franz Lemmermeyer

Dedicated to Dr. Phil. Joh. Eduard Böttcher Teacher at the Realgymnasium in Leipzig With Thankfulness and Admiration. Oswald Baumgart, 1885

Contents

Part I

Presentation of the Proofs of the Quadratic Reciprocity Law

1

From Fermat to Legendre . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3

2

Gauss’s Proof by Mathematical Induction. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7

3

Proof by Reduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1 Gauss’s Third Proof [26] . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2 Gauss’s Fifth Proof [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3 Eisenstein’s Geometric Proof [17] . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4 Proof by Genocchi [37] . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5 Proof by Stern [70] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6 Proof by Zeller [73] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7 Proof by Kronecker [43] . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8 Proof by Bouniakowski [3] . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9 Proof by Schering [67, 68] . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10 Proof by Petersen [63, 64] . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11 Proof by Voigt [71] .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12 Proof by Busche [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

15 15 17 19 20 22 24 26 28 31 33 34 35

4

Eisenstein’s Proof Using Complex Analysis . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

41

5

Proofs Using Results from Cyclotomy .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1 Proof by Gauss (7th Proof) [34]Gauss and Lebesgue (2nd Proof) [52] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2 Gauss’ Fourth Proof [27]. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3 Gauss’ Sixth Proof [29] . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4 Proof by Cauchy [8],Cauchy Jacobi [56, p. 391], Jacobi Eisenstein [15] .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5 Second Proof by Eisenstein [14]. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6 Proof by Liouville [61] .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7 First Proof by Lebesgue [50] . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

45 45 48 52 55 56 60 60 ix

x

Contents

6

Proofs Based on the Theory of Quadratic Forms . . .. . . . . . . . . . . . . . . . . . . . 1 Preliminary Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Gauss’s Second Proof [25, art. 257], [12, Suppl. IV, X].. . . . . . . 1.2 Kummer’s First Proof [48] .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Kummer’s Second Proof [48]. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7

The Supplementary Laws of the Quadratic Reciprocity Law and the Generalized Reciprocity Law . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1 The Supplementary Laws . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Proof of Formula (I) Using “Associate Residues” (See [21, p. 135], [25, Art. 109]), for Formula (II) Using Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Proof of the Supplementary Laws Using Reduction . . . . . . . . . . . 1.3 Proof of the Second Supplementary Law Using Cyclotomy .. . 1.4 Proof of the Supplementary Laws Using Quadratic Forms [25, art. 262] . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2 The Generalized Reciprocity Law .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

8

Algorithms for Determining the Quadratic Character .. . . . . . . . . . . . . . . . 1 Gauss’s Method for Determining (a/b) [32, p. 59]. . . . . . . . . . . . . . . . . . . . 2 The Algorithms by Eisenstein [13] and Lebesgue [51] .. . . . . . . . . . . . . . 2.1 Eisenstein .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 The Algorithms of Lebesgue . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3 The Algorithms by GEGENBAUER [35] .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 The Second Method .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4 An Algorithm by Kronecker . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part II

9

63 63 64 65 68 71 71

71 72 73 73 74 77 77 79 79 79 81 81 82

Comparative Presentation of the Principles on Which the Proofs of the Quadratic Reciprocity Law Are Based

Gauss’s Proof by Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

85

10 Proofs by Reduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

89

11 Eisenstein’s Proofs Using Complex Analysis . . . . . . . .. . . . . . . . . . . . . . . . . . . . 107 12 Proofs Using Results from Cyclotomy .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 111 13 Proofs Based on the Theory of Quadratic Forms . . .. . . . . . . . . . . . . . . . . . . . 125 14 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 127 15 Proofs of the Quadratic Reciprocity Law . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131

Contents

xi

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 163 Author Index.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 167 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 171

Introduction

Higher arithmetic in essence divides into two main parts, the theory of congruences and the theory of homogeneous forms.2 An integral part of the theory of congruences is formed by the theory of binomial3 congruences, whose pivotal point is the theory of power residues. “The reciprocity laws are the cornerstone of the latter theory” (see [47, p. 19]). Although finding these laws “of simply stated content” (see [31]) using induction4 was comparatively easy, finding their proofs was connected with huge difficulties: to this end new methods had to be found, and proofs had to be sought in areas that seemed to have no connection with the number theory. And yet at first it was only possible to verify the correctness of the quadratic reciprocity law. But the principles found in the different proofs of the quadratic reciprocity law could be generalized to such a degree that they also could be used for deriving the general reciprocity laws. In the following we will present all known proofs of the quadratic reciprocity law and carefully compare the principles on which they are based. The author believes that such a first comparison is not completely useless, because this law is the fundamental result in the theory of quadratic residues and nonresidues, moreover because the principles on which they are built allow us to find new and very general methods, and finally because the proofs of this law have induced a very welcome interaction between some, until then almost or completely isolated, areas

2

[FL] Jacobi also divided the number theory into these two areas, as did H.J. Smith in his report; notably absent from this list is cyclotomy, which both authors apparently subsumed into the theory of congruences.

3

[FL] In modern terms: polynomial equations in two variables over residue class rings, i.e., solutions of f .X; Y / D 0 over finite fields

4

[FL] Clearly he is not talking about mathematical induction; finding something by induction in those days simply meant making a conjecture based on empirical evidence. xiii

xiv

Introduction

of mathematics. In addition, the history of this theorem is a faithful reflection of the simultaneous history of mathematics in the small.5 It was Professor SCHEIBNER who notified me of this remarkable and attractive fact. I would like to thank my highly respected teacher for this as well as for the support and the stimulation I have received from him. In the first part I have presented all known proofs, as far as they were accessible to me, ordered in chapters in such a way that the proofs of one chapter are based on the same main idea. Within each chapter, the proofs are given in chronological order. The principles themselves are developed in the second part. Historical notes are given at the beginning and the end of this thesis. For the convenience of the reader, and also in order to highlight the similarities and the differences of the proofs, I have tried to choose a common notation and presentation. It goes without saying that not only the main idea but also the individual structure of the different proofs has remained untouched.

5

[FL] This seems to be an allusion to Haeckel’s “Biogenetic Law” first published in 1866, according to which ontogeny recapitulates phylogeny, i.e., an embryo repeats in its development the evolutionary history of its species as it passes through stages in which it resembles its remote ancestors.

Part I

Presentation of the Proofs of the Quadratic Reciprocity Law

Chapter 1

From Fermat to Legendre

After BACHET DE M ÉZIRIAC [1] had brought the theory of linear diophantine equations to a certain closure, mathematicians were faced with the question of solving equations of the second degree, in particular the binomial congruence of degree 2. In other words, the problem was to find simple conditions for the solvability of the congruence x 2  p mod q; where p and q are given integers. At first only special cases were investigated. It transpires from a letter of FERMAT to the Englishman KENELM DIGBY (see [72, vol. II, p. 857]) in 1658 that already FERMAT knew the conditions for which odd primes q the numbers ˙1, 2, ˙3, 5 are quadratic residues or nonresidues; it is also clear from a letter of FRENICLE (see [23, p. 168]) to FERMAT in 1641 that already FRENICLE knew when 2 is quadratic residue or nonresidue of some prime number. Probably, however, as also LAGRANGE [49, p. 337] assumes, this had been known to FERMAT before and was FRÉNICLE’s answer to a question of FERMAT. All these results were found by induction and given without proofs. For 1, the theorem was first proved by EULER [21, p. 135] (see also p. 71 of this report) using associated residues (residua socia); his method did not succeed for ˙2. This gap was closed by LAGRANGE [49, p. 349, 351]. It is a remarkable fact that EULER did not succeed in finding the proof for ˙2, because he actually knew (see [19]) the proof of the law for ˙3. As for ˙5, it was again LAGRANGE [49, p. 352] who first succeeded in proving under which conditions this number is a quadratic residue or nonresidue of a given prime number. These results, which were without influence on the actual presentation of the law, are presented for the sake of completeness and in order to make clear with which difficulties mathematicians had to deal here. Although it cannot be denied © Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_1

3

4

1 From Fermat to Legendre

that the invention of infinitesimal calculus has essentially diverted mathematicians from number theory, it is nevertheless a remarkable fact that simple results such as those mentioned above could remain without proofs for more than 100 years. Until now, only special cases were treated. The first to try to recognize and formulate our law in its full generality was EULER. And he succeeded in making an important step forward. In [20] he gives four theorems that completely contain the quadratic reciprocity law. They read1 : 1. Si divisor primus fuerit formae 4ns C .2x C 1/2 , existente s numero primo, tum in residuis occurent numeri Cs et s. 2. Si divisor primus fuerit formae 4ns  .2x C 1/2 , existente s numero primo, tum in residuis occuret numerus Cs, at s in non-residuis. 3. Si divisor primus fuerit formae 4ns  4z  1 excludendo omnes valores in forma 4ns  .2x C 1/2 contentos, existente s numero primo, tum in residuis occuret numerus s; at s erit non-residuum. 4. Si divisor primus fuerit formae 4ns C 4z C 1 excludendo omnes valores in forma 4ns C .2x C 1/2 contentos, existente s numero primo, tum tam Cs quam s in non-residuis occuret. As a simple calculation shows, Euler has made a mistake in the formulation of 3. Indeed, the second part of this theorem has to be stated as follows: If s has the form 4n C 1, then Cs is nonresidue and s residue; for s D 4n  1, the opposite happens.2 These four theorems, also stated without proof, completely contain, as we have already remarked and as will be shown later on without effort by comparison, the quadratic reciprocity law. GAUSS seems to have been unaware of EULER’s article just discussed and therefore credits L EGENDRE (see [25, Art. 151]) with the discovery of our law. It was this famous number theorist, however, who for the first time clearly and explicitly enunciated the fundamental theorem using formulas, namely 1785 in [53, pp. 516–517], and who gave a partial proof. In the fourth section of the work just mentioned he stated eight theorems, in which A, a are primes of the form 4n C 1,

1

[FL] Using modern notation, this can be translated as follows:

1. A prime has the form p D 4ns C .2x C 1/2 if and only if p  1 mod 4 and p  y 2 mod s. Thus the first statement claims that if p  1 mod 4 is prime and p  x 2 mod s for some prime p s, then ˙s  y 2 mod p. In other words: if p  1 mod 4 then . s / D C1 ) . ˙s / D C1. p 2 2. If p  3 mod 4 is prime and p  x mod s for some prime s, then s  y 2 mod p and s 6 y 2 mod p. In other words: if p  3 mod 4 then . p / D C1 ) . ps / D C1; . s / D 1. s p 2 3. If p  3 mod 4 is prime and p 6 x mod s for some prime s, then s  y 2 mod p and p / D C1; . ps / D 1. s 6 y 2 mod p. In other words: if p  3 mod 4 then . s / D 1 ) . s p 4. If p  1 mod 4 is prime and p 6 x 2 mod s for some prime s, then ˙s 6 y 2 mod p. In other p / D 1. words: if p  1 mod 4 then . s / D 1 ) . ˙s p 2

[FL] It seems that Baumgart is wrong here. I cannot see anything wrong with Euler’s formulation.

1 From Fermat to Legendre

5

B and b, on the other hand, primes of the form 4n C 33 : Théorème I. Théorème II. Théorème III. Théorème IV. Théorème V. Théorème VI. Théorème VII. Théorème VIII.

a1

Si b 2 b1 Si a 2 A1 Si a 2 A1 Si a 2 b1 Si a 2 a1 Si b 2 B1 Si b 2 B1 Si b 2

b1

D C1; il s’ensuit a 2 a1 D 1; il s’ensuit b 2 a1 D C1; il s’ensuit A 2 a1 D 1; il s’ensuit A 2 a1 D C1; il s’ensuit b 2 b1 D 1; il s’ensuit a 2 b1 D C1; il s’ensuit B 2 b1 D 1; il s’ensuit B 2

D C1: D 1: D C1: D 1: D C1: D 1: D 1: D C1:

Thus it can be seen that LEGENDRE used FERMAT’s Theorem for stating his theorems. In fact it follows from x 2  p mod q

and p q1  1 mod q;

that the solvability of the congruence x 2  p mod q depends on p .q1/=2 . In fact, if p .q1/=2  1 mod q, then this congruence is solvable; if, however, p .q1/=2  1 mod q (other cases never occur), then this congruence is not solvable. LEGENDRE also was the first to give the theorem in the form in which we currently state the theorem, namely in [54]. On p. 186 LEGENDRE remarks at first: Comme les quantités analogues N .c1/=2 se rencontreront fréquemment dans le cours de nos recherches, nous emploierons le caractère abrégé . Nc / pour exprimer le reste que donne N .c1/=2 divisé par c, reste qui suivant ce qu’on vient de voir ne peut être que C1 ou 1.4

On p. 214 he writes Quels que soient les nombres premiers m et n, s’ils ne sont pas tous deux de la forme 4x 1, on aura toujours . mn / D . mn /; et s’ils sont les deux de la forme 4x 1, on aura . mn / D . mn /. Ces deux cas géneraux sont compris dans la formule n m

3 .a1/=2

b

D .1/

n1 m1 2  2

m n

:5

D 1, . . . should actually read b .a1/=2  1 mod a . . . .

[FL] Since the analogous quantities N .c1/=2 will occur often in our researches, we shall employ the abbreviation . Nc / for expressing the residue that N .c1/=2 gives upon division by c, and which, according to what we just have seen, only assumes the values C1 or 1.

4

[FL] Whatever the prime numbers m and n are, if they are not both of the form 4x 1, one always has . mn / D . mn /; and if both are of the form 4x  1, one has . mn / D . mn /. These two general cases are contained in the formula n m n1 m1 D .1/ 2  2 : m n

5

6

1 From Fermat to Legendre

LEGENDRE calls this law the quadratic reciprocity law, whereas GAUSS called it theorema fundamentale in doctrina de residuis quadraticis. Thus 150 years after the first special cases had been discovered one of the most important number theorists succeeded in stating the theorem in its most general form and in its most elegant way. Later on we will have the opportunity to return to L EGENDRE’s attempts to prove the theorem. Here we only remark that his proof is incomplete; and precisely because of this incompleteness we will not discuss it here (see [25, Art. 151, 296, 297 and Additamenta].) Applying LEGENDRE ’s notation we have now observed:  1 

.I /

q 2

.II/

q  p  q 

.III/

q

p

D .1/

q1 2

D .1/

q 2 1 8

D .1/

p1 q1 2  2

I I ;

where p and q denote positive odd primes. These three formulas make up the quadratic reciprocity law. In the following five chapters we will give proofs mainly for formula (III); the supplementary laws of the quadratic reciprocity law, as the laws expressed by formulas (I) and (II) are called, will be discussed in a special chapter on p. 71. Before we do this, we have to introduce a generalization of LEGENDRE ’s symbol given by JACOBI, because it is of great importance for computing6 the Legendre symbol. In fact, whereas LEGENDRE assumes that the q in . aq / is an odd positive prime and a an integer coprime to q, JACOBI allows composite numbers q D b. In . ab /, a and b are assumed to be coprime integers, not both negative, and b odd. These generalized Legendre symbols were defined by JACOBI using the formulas a   a  a  a  D  ; pqr    p q r  a  a D ; b b  abc      a  b  c  D  ; q q q q 

where p, q, r, . . . denote primes that may be distinct or not.

6

[FL] Actually, it also is important for GAUSS’s first proof in the form given by D IRICHLET, which will be discussed in the next section.

Chapter 2

Gauss’s Proof by Mathematical Induction

1. GAUSS distinguishes in his first proof, just as L EGENDRE, eight different cases according to the different nature of the primes in question, so that the actual proof is seperated into eight proofs. The eight individual cases are: 1. 2. 3. 4. 5. 6. 7. 8.

If q D 4n C 1, p D 4n C 1 and . pq / D 1, then we have to prove that . pq / D 1; If q D 4n C 1, p D 4n C 3 and . pq / D 1, then we have to prove that . pq / D 1; If q D 4n C 1, p D 4n C 1 and . pq / D 1, then we have to prove that . pq / D 1; If q D 4n C 1, p D 4n C 3 and . pq / D 1, then we have to prove that . pq / D 1; If q D 4n C 3, p D 4n C 3 and . pq / D 1, then we have to prove that . pq / D 1; If q D 4n C 3, p D 4n C 1 and . pq / D 1, then we have to prove that . pq / D 1; If q D 4n C 3, p D 4n C 3 and . pq / D 1, then we have to prove that . pq / D 1; If q D 4n C 3, p D 4n C 1 and . pq / D 1, then we have to prove that . pq / D 1.

These eight theorems give the complete reciprocity law1 ; they can be collected into the following three: (I) If q D 4n C 1 and . pq / D 1, then we have to prove that . pq / D 1; (II) If q D 4n C 1 and . pq / D 1, then we have to prove that . pq / D 1; (III) If q D 4n C 3 and . ˛q / D 1, then we have to prove that . ˛q / D .1/.˛1/=2 . q1

In case (III) we have ˛ D ˙p. In fact, if . pq / D 1, then . pq /  p 2 mod q implies . p / D C1, hence the case . pq / D 1 does not have to be investigated. q Combining cases (I) and (III) our proof is reduced to showing that if 1. q D 4n C 1, 4n C 3 and . ˛q / D 1, then . ˛q / D .1/

˛1 q1 2 2

, and

1

From now on, we shall omit the word “quadratic” from quadratic residue, nonresidue, reciprocity law if it is clear what we are talking about. © Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_2

7

8

2 Gauss’s Proof by Mathematical Induction

2. q D 4n C 1 and . pq / D 1, then . pq / D 1. Here p represents an arbitrary odd positive prime, ˛ an arbitrary odd positive or negative prime. In the following we may and will assume that q > p and suppose that the reciprocity law holds for all primes less than q and for all composite integers coprime to p that are products of these primes. ˛1 q1 2. If . ˛q / D C1, then we have to show that . ˛q / D .1/ 2 2 . The assumption implies that the congruence x 2  ˛ mod q is solvable. Let e denote its even solution less than q; then e 2 D ˛ C fq:

(2.1)

Here f is a nonzero integer, because otherwise ˛ D e 2 would be even; moreover, f is positive because otherwise we would have ˛ D Cp and p e 2 > q, contradicting the assumption that q > p; finally f is odd because fq D e 2  ˛ is odd. Moreover f < q  1 because e and p are smaller than q  1, which implies qf D e 2 C ˛ < .q  1/2 C .q  1/ D q.q  1/, hence f < q  1. Now there are two possibilities in Eq. (2.1). 1. e and f are coprime to ˛. From e 2  fq mod ˛ it follows that . fq˛ / D 1, or . f˛ / D . ˛q /, while e 2  ˛ mod f implies . f˛ / D 1. Thus we get q ˛

D

f  ˛

D

˛ f

.1/

˛1 f 1 2 2

D .1/

˛1 f 1 2 2

;

since by induction assumption the reciprocity law holds for all primes < q. Now e  0 mod 2, hence ˛  qf mod 4 or .˛ C 1/  qf  1  q  1 C f  1 mod 4 and  ˛2  1  q1 ˛1 f 1 ˛1   C  mod 2:  4 2 2 2 2 But ˛ 41 D ˛1  2 which implies 2

˛C1 2

is the product of two consecutive integers, hence even,

q1 ˛1 f 1 ˛1    mod 2; 2 2 2 2 and this is what we wanted to prove. 2. f and e are divisible by ˛. If f D ˛ and e D ˛", then ˛"2 D 1 C q;

(2.2)

2 Gauss’s Proof by Mathematical Induction

9

where ˛ and q are coprime. Now . ˛ / D 1, and 12  q mod ˛ implies . ˛q / D

.  /, hence using our induction assumption we get ˛ q  ˛

D

   ˛  ˛



D .1/

˛1 C1 2 2

:

Since e is even we get q  1 mod 4, and this implies  C 2  q mod 4, ˛1 q1 or C1  q1 mod 2. Thus . ˛q / D .1/ 2 2 , and this is what we wanted to 2 2 prove. 3. If . pq / D 1 and q D 4n C 1, then we have to show that . pq / D 1. GAUSS first proves the theorem that for any prime q D 4nC1 there always exists a prime p0 < q such that q is a quadratic nonresidue modulo p 0 , and distinguishes two cases. 1. q D 8n C 5. If q  2 D p 0 is prime, then q  2 mod p 0 and therefore . pq0 / D . p20 / D .1/Œ.8n3/ 1=8 D 1.2 Thus the prime q  2 D p 0 has the desired property. If q  2 is composite, however, then at least one prime factor of q  2 must have the form 8n˙3. In fact, if q2 was a product of primes 8n˙1, then we would have q  2 D 8 ˙ 1,3 contradicting the assumption q D 8n C 5. Denoting such a prime factor of q  2 of the form 8n ˙ 3 by p 0 , then q  2 mod p 0 and again .q=p 0 / D 1, hence in this case too there is a prime p 0 < q of which q is a quadratic nonresidue. 2. q D 8nC1. Assume that q is a quadratic residue modulo all odd primes  2mC1 (< q); then q  1 mod 8, hence q is a quadratic residue modulo all integers whose prime factors all are  2m C 1. Thus there exist integers k satisfying the congruence 2

k 2  q mod M;

M D 1  2    .2m C 1/ D .2m C 1/Š

Clearly k is coprime to M and q since otherwise M and q would have a factor in common, which is impossible. This congruence implies .k 2  1/    .k 2  m2 /  .q  12 /.q  22 /    .q  m2 / mod M: Now k  .k 2  1/    .k 2  m2 /  .k  m/.k  m C 1/    k.k C 1/    .k C m/ mod M;

2

The proof of the formula .2=p/ D .1/.p

3

[FL] The original reads: q  2 D 8  1.

2 1/=8

can be found on p. 71 ff.

10

2 Gauss’s Proof by Mathematical Induction

as a product of 2m C 1 consecutive integers, must be divisible by M , hence k.q  12 /.q  22 /    .q  m2 / Dz 1  2    .2m C 1/

(2.3)

is an integer. But now4 .2m C 1/Š D Œ.m C 1/  m  Œ.m C 1/  .m  1/    Œ.m C 1/  0  Œ.m C 1/ C m    Œ.m C 1/ C 1 D .m C 1/Œ.m C 1/2  m2     Œ.m C 1/2  11 : Plugging this into (2.3) we find that zD

q  12 q  m2 1   2 m C 1 .m C 1/  1 .m C 1/2  m2

p is an integer. Now let m be the maximal integer less than q; then our assumption that 2m C 1 < q still holds, and since .m C 1/2 < q we find that z is a proper fraction. Our assumption on the residue character of q is therefore false, and we have proved the following result: if q is a prime of the form 8n C 1, then p there exists an odd prime p0 < 2 q C 1, hence < q, of which q is a quadratic nonresidue. 4. Thus for every odd prime q D 4n C 1 there is an odd prime p 0 < q such that 0 0 . pq0 / D 1. But then we must have . pq / D 1: in fact, if . pq / D C1, then by the p 0 1 q1

preceding we would have . pq0 / D .1/ 2  2 D C1. Thus the reciprocity law holds for p 0 and q. We have to show that . pq / D 1. Since . pq0 / D 1, it is sufficient to prove 0

that . ppq 0 / D C1. By assumption we have . pq / D 1, hence . ppq / D C1. Thus the congruence x 2  pp0 mod q is solvable. Let e denote its even solution < q; then e 2 D pp0 C fq;

(2.4)

where f is an odd integer less than q. Now we have to distinguish: 0

/ D 1, 1. e and f are not divisible by p or p 0 . Then e 2  pp0 mod f implies . pp f q f qf and e 2  qf mod pp0 implies . pp 0 / D 1 or . pp0 / D . pp0 /. Now we get

 q  f 1 pp0 1 2  2 : D .1/ pp0

4

[FL] Baumgart does not explain the disappearance of k from the numerator in (2.3) below. Of course, since k is coprime to M D .2m C 1/Š we see that z is an integer if and only if z=k is one.

2 Gauss’s Proof by Mathematical Induction

11

Since e  0 mod 2 and q  1 mod 4 we have f  pp0 mod 4, hence f  1 pp0  1 pp0 C 1 pp0  1   mod 2: 2 2 2 2 The right-hand side of this congruence is the product of two consecutive integers, hence even, and this implies  q  D 1; pp0 which is what we wanted to prove. 2. e and f are divisible by p 0 but not by p. Setting e D "p 0 and f D p 0 , then "2 p 0 D p C q, where  is coprime to p, p 0 and q. Thus we get  pp0  

D

 qp 0  p

D 1 or

q p

D

   p 0  : p p

Since moreover "2 pp0 D p 2 C pq, we have . qp / D 1, hence . q / D . p /. p0 p0 p0 This implies  q     pp0  p 0  p  D : pp0 pp0  p p0 Thus using our induction assumption we get  q  pp0 1 1 p1 p 0 1 p 0 1 2 2 C 2 2 C 2 : D .1/ pp0 But now " in "2 p 0 D p C q is even since e  0 mod 2, hence   p mod 4, hence   1 pp0  1 p C 1 p0  1 p C 1 n pp0  1 p0  1 o  C   C 2 2 2 2 2 2 2 0 pC1 p 1  mod 2;  p  2 2 and this implies . ppq 0 / D C1, which is what we wanted to prove. 3. The case where e and f are divisible by p but not by p 0 is analogous to the preceding one. 4. e and f are divisible by p and p 0 . We write e D "pp0 , f D pp0 ; then ."pp0 /2  pp0 C qpp0 where  is coprime to p, p0 , and q.

and "2 pp0 D 1 C q;

12

2 Gauss’s Proof by Mathematical Induction 0

This implies . q / D C1, or . ppq 0 / D .  /. Moreover . pp / D 1, hence pp0 pp0 0

. ppq 0 / D .  /. pp /, and this equation leads to pp0

 q  C1 pp0 1 2  2 D .1/ pp0 because , p and p 0 are all smaller than q: thus we may apply our induction hypothesis. But now e is even and q  1 mod 4, hence   1 mod 4 and C1  0 mod 2; thus we get 2  q  D C1: pp0 This completes the second part of the proof. We have proved the reciprocity law for p and q only under the assumption that there exist two primes, smaller than the maximum of p and q, for which the law holds, and that if the reciprocity law holds for primes, then the corresponding law also holds for general residue characters. The first assumption poses no problem because the two smallest odd primes satisfy EULER’s law. Concerning the second assumption we remark the following (see GAUSS [25, Art. 132] and DIRICHLET [11, p. 143]). If P and Q are positive composite odd coprime integers, and if we factor P and Q into primes, P D p  p 0  p 00   

Q D q  q 0  q 00    ;

then we get  P  Q  Y  p  q  D ; Q P q p where each p has to be combined with each q. Assuming that the quadratic reciprocity law holds for all p and q, we get  P  Q  Y P p1 q1 p1 q1 D .1/ 2  2 D .1/ 2  2 ; Q P where the sum is over all pairs p and q, which implies X p1 X q1 Xp1 q1  D  : 2 2 2 2 From RD

Y

rD

Y X Œ.r  1/ C 1  1 C .r  1/ mod 4

2 Gauss’s Proof by Mathematical Induction

for odd integers r we find that reason we find

R1 2

and

13

P r1 2

have the same parity. By the same

X p  1 q  1 X p1 X q1 P 1 Q1 mod 2;      2 2 2 2 2 2 and this gives  P  Q  P 1 Q1 D .1/ 2  2 : Q P The assumption we made is therefore true, and thus the quadratic reciprocity law holds in full generality.

Chapter 3

Proof by Reduction

1 Gauss’s Third Proof [26] 1. If q denotes a prime number, then 1; 2; : : : ; q1 is a complete system of 2 incongruent positive minimal1 residues modulo q; on the other hand, if a is coprime to q, then a, 2a, . . . , q1 a is a system of q1 incongruent residues which do not 2 2 necessarily form a half-system modulo q. If, in this last set, 1 , . . . ,  are the positive and 1 , . . . ,  the negative minimal residues modulo q, then we can observe that the  and  are nonzero and pairwise distinct, hence congruent modulo q in some order to the numbers 1; 2; : : : ; q1 . 2 In fact, if two residues ta mod q and sa mod q were equal, then we would have .t  s/a  0 mod q; or, neglecting the signs of the , .t C s/a  0 mod q; and both of these are impossible since t and s are distinct and smaller than q2 . Thus we get a

q1 2

12

Y Y q1   mod q;  .1/ 2

i.e., a

q1 2



a q

D .1/ mod q:

[FL] A minimal residue system modulo p consists of the numbers 0, ˙1, ˙2, . . . , ˙ p1 ; among 2 p1 these, the residues 1; 2; : : : ; 2 are called “positive”. Occasionally we will call this a “half-system modulo p” in the following. 1

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_3

15

16

3 Proof by Reduction

This formula holds for every integer a coprime to q, in particular for some prime number p distinct from q, and we get the result: If p and q are positive odd primes, then . pq / D .1/, where  denotes the number of negative minimal residues in p, p mod q. 2p, . . . , q1 2 2. In order to investigate the number  found in the preceding lemma more closely, GAUSS denotes by Πxy  the biggest integer  yx and finds q1

D

2 nh X 2pr i

q

rD1

2

h pr io q



X h 2pr i r

q

mod 2:

 terms according to the formula Œp  x C In the last sum we change the last Œ qC1 4 Œx D p  1, that is, using h 2pr i

D p1

q

h p.q  2r/ i q

;

and omitting multiples of p  1  0 mod 2 we get 

h 2p i q 

hp i q

h 4p i

C

q



h 3p i q

C ::: C  :::

h q  1  2ΠqC1  i 4

q

h ΠqC1 p i 4

q

;

hence q1



2 h X pr i

rD1

q

mod 2:

terms, whose first term vanishes if we assume that p < q, This sum consists of q1 2 and whose last term equals p1 because of 2 h q1 p i 2

q

h q p1 C 2 D q

qp 2

i :

Thus some terms must occur more than once. .mC1/p Now if Πmp  D n, then the inequality q  D n  1 and Πq mC1 mp p denote two distinct positive odd primes, there are exactly q integers congruent modulo p to some positive minimal residue rp , namely rp ; rp C p; : : : ; rp C .q  1/p:

(3.2)

This set of integers forms a complete residue system modulo q, because the difference of any two of its elements is not divisible by q. If we reduce the q

18

3 Proof by Reduction

elements of (3.2) modulo q, then there are q1 positive minimal residues, q1 2 2 negative minimal residues, and the remaining term is a multiple of q. We have proved: Theorem 3.1 In the sequence 1; 2; : : : ; pq  1 there are p1  q1 integers that are 2 2 congruent to positive minimal residues modulo p and modulo q, and there are also p1 q1  2 integers that are congruent to positive minimal residues modulo p and to 2 negative minimal residues modulo q. 2. Let us introduce the following notation: pq  1 ; 2 p1 ; rp D 1; 2; : : : ; 2 q1 ; rq D 1; 2; : : : ; 2

pq C 1 ; : : : ; pq  1; 2 pC1 Rp D ; : : : ; p  1; 2 qC1 Rq D ; : : : ; q  1; 2

s D 1; 2; : : : ;

SD

and let .s/rR denote the number of integers s that are congruent to positive minimal residues modulo p and to negative minimal residues modulo q; moreover let .s/Rr denote the number of integers s that are congruent to negative minimal residues modulo p and to positive minimal residues modulo q. Then we have .s/Rr D .S /rR ;

(3.3)

where .S / has the same meaning with respect to S as .s/ has with respect to s. According to Theorem 3.1 we have .s/Rr C .S /rR D

p1 q1  : 2 2

(3.4)

G AUSS now considers the sequence

q

qC1 ; 2 qC1 C 2

::: C

p3 q 2

qC1 2

: : : q  1; : : : 2q  1; ::: ::: : : : p1 q  1: 2

 q1 , and all of them are congruent to negative The number of terms is p1 2 2 minimal residues modulo q because the next row would start with pqC1 . GAUSS 2 q1 now partitions these p1  integers into three classes. 2 2 1. The first class contains the terms that are congruent modulo p to positive minimal residues; there are .s/rR of them.

3 Eisenstein’s Geometric Proof [17]

19

2. The second class contains the terms that are congruent modulo p to negative minimal residues; there are .s/RR of them. 3. The remaining terms have the form prq  Rq mod q; their cardinality  is equal to the exponent belonging to the residue p and the modulus q in GAUSS’s Lemma. This partition yields the equality p1 q1  : 2 2

.s/rR C .s/RR C  D

(3.5)

Similarly, considering the system of integers

p

pC1 ; 2 pC1 C 2

::: q3 p C 2

p

pC3 ; 2 pC3 C 2 ;

::: C

pC1 q3 p 2 2

pC3 2

: : : p  1; : : : 2p  1; ::: ::: : : : q1 p1 2

and letting  denote the exponent belonging to the residue q and the modulus p in GAUSS’s Lemma, we get .s/Rr C .s/RR C  D

p1 q1  ; 2 2

.S /rR C .s/RR C  D

p1 q1  : 2 2

or, using (3.3), (3.6)

Adding (3.5) and (3.6) and subtracting (3.4) from the result gives 2.s/RR C  C  D

p1 q1  ; 2 2

hence C 

p1 q1  mod 2; 2 2

which is what we wanted to prove.

3 Eisenstein’s Geometric Proof [17] In a cartesian coordinate system, EISENSTEIN draws parallels with distance 1 to the axes and calls the points of intersection of these lines lattice points.

20

3 Proof by Reduction B

Fig. 3.1 Lattice Points in Eisenstein’s Proof

C

O

Now consider a line given by the equation y D

p q x,

F

D

A

where p and q as usual are

, y attains the two distinct positive odd primes. Now if, for x D a D 1; 2; : : : ; p1 2 value b, i.e., if b D q pa , then Œb D Œ qa  denotes the number of lattice points on the p line orthogonal to the x-axis between y D 0 and y D b. According to GAUSS we have q p

D .1/

for  D

hqi p

C ::: C

h p1 q i 2

p

:

Now it is clear by looking at Fig. 3.1 that  is equal to the number of lattice points . Similarly, using the same equation inside the triangle OAD, where OA D p1 2 written in the form x D pq y, we get for the exponent  in . pq / D .1/ , which (again up to some even term) is equal to the number of lattice points within OBCD, where OB D q1 . Thus  C  is congruent modulo 2 to the number of lattice points 2 inside OABC ; in fact, no point occurs twice since these clearly would have to lie on the line itself, which is not possible. The number of such lattice points now is obviously equal to p1  q1 , and we have given yet another proof of the reciprocity 2 2 law.

4 Proof by Genocchi [37] Assume that p and q satisfy the usual conditions and put rD where h
r, which is impossible. , v will attain Thus as k runs through the integers 1, 2, . . . , q1 2 ( i positive values if h0 < p=2, positive values if h0 > p=2,

i C1 hence2 k

k

card: pos: v D card: pos: u C 1; k

p ; 2 p h0 > : 2

h0
p2 ; then we have   card: pos: v  card: pos: u mod 2: h;k

h;k

(3.10)

Similarly, if card. pos. u0 denotes the number of positive .kp  hq/ and  the number of residues kp that are congruent to negative minimal residues modulo q, then   card: pos: v  card: pos: u0 mod 2; h;k

h;k

(3.11)

and this implies when we plug in the values for u and u0 : card.h;k pos. .hq  kp/  card.h;k pos. .kp  hq/   C  mod 2; which is what we needed to prove.

2

Here the notation card:k pos: u is taken from SCHERING and denotes the number of positive values of v as k runs through the half-system.

22

3 Proof by Reduction

5 Proof by Stern [70] This proof is not completely correct. Nevertheless it is listed here for the sake of completeness and because it introduces a fruitful new idea.3 If p and q as before denote two odd prime numbers, then we have the congruence 12

p1  .1/u  2  4    .p  1/ mod p; 2

where u denotes the number of odd integers in the interval 1, . . . , uD

(3.12)

p1 . Thus we 2

p" ; 4

have

(3.13)

where " is equal to 1 or 1 according as p has the form 4n C 1 or 4n  1. Similarly we have q  2q   

p1 p1 q D q .p1/=2  1     .1/u1  2  4    .p  1/ mod p; 2 2

where u1 denotes the number of positive odd residues modulo p among the numbers q, 2q, . . . , p1 q, hence 2 q

p1 2

12

p1 p1  q 2 .1/u 2  4    .p  1/ mod p; 2

which implies q

p1 2



q p

D .1/u1 u :

Thus we get  p  q  q

p

D .1/u1 Cv1 uv ;

(3.14)

where v and v1 have the same meaning with respect to q as u and u1 with respect to p. Thus in order to prove the reciprocity law, in light of (3.13) we only have to show that4 ( p1 C q1 mod 2 if p and q are similar; 4 4 u1 C v1  p1 (3.15) mod 2 if p and q are not similar: 4

3

[FL] Gegenbauer (see his article referred to in the appendix) observed that Stern’s article does indeed contain a complete and correct proof of the quadratic reciprocity law. Here two integers are called similar if they both have the form 4n C 1 or 4n C 3, and not similar if one has the form 4n C 1 and the other 4n C 3.

4

5 Proof by Stern [70]

23

Now let u0 and v0 denote the number of even positive residues among q; 2q; : : : ;

p1 q mod p; 2

p; 2p; : : : ;

q1 p mod q; 2

and

respectively. Then we have u1 C u0 D

p1 , v1 2

u 1 C v1 C u 0 C v0 D

C v0 D

q1 , 2

and

p1 q1 C : 2 2

Now we introduce the abbreviations G D u 0 C v0 I

U D u 1 C v1 ;

(3.16)

then in order to prove the reciprocity law we have to show, using (3.15), that U  G  0 mod 4 if p and q are similar; U  G  1 mod 4 if p and q are not similar; or, in other words, that5 U G 

qp mod 4: 2

(3.17)

In order to verify the formula U  G  qp mod 2, STERN observes that some 2 residue r cannot occur both as xq mod q and yp mod q. In fact, if we had aq D gp C r and a0 p D g 0 q C r, then we would get .a C g 0 /q D .a0 C g/p:

(3.18)

But now a0 < q2 , hence g0 < p2 , and since we also have a < p2 we find a C g 0 < p. Recalling the properties of p and q this implies the impossibility of Eq. (3.18) and thus the validity of the above claim. Now we make the assumption q > p and set aq D gp C r;

a
p2 . 2. If  q1 p q kp  r mod q; k< ; and p, this 1. If k D 0, then k 0 D q1 2 2 2 residue is positive and therefore does not contribute to . 2. If k D k 0 D q1 , then there only is a residue if q has the form 4n C 1. 4 Thus if q D 4n C 3 we get  0 mod 2 and  C   If q D 4n C 1, then kp D

q1 p 4



p˙q 2

p1 mod 2: 2

mod q and we have to distinguish:

1. p  1 mod 4: then kp gives a positive residue, and  0 mod 2. 2. p  3 mod 4: here q has to be taken negative, and we get  1 mod 2. Combining these cases we get our well-known formula.

26

3 Proof by Reduction

7 Proof by Kronecker [43] Let p and q again denote two distinct odd primes, and define . pq / as the sign of p1 q1

2 Y 2  Y h

p

hD1 kD1

then it is immediately clear that  p  q  q

p

Moreover it follows from Yh



p

h;k

that



D .1/

k I q

p1 q1 2  2

:

(3.23)

1 Y kp  k D h q p q h;k

p q

D .1/

P

kp kΠq 

:

0

 mod q, and therefore If we assume p 0  p mod q, then Πkpq   Πkp q  p0  q

D

p  I q

(3.24)

similarly if p 0  p mod q, then we get  p0  q

D .1/

q1 2

p : q

(3.25)

Finally if kp  ˙k 0 mod q, where k 0 belongs to a minimal half-system modulo q, and if r denotes the positive integer k 0 or q  k 0 , then h kpp0 i q

D p0

h kp i q

C

h p0 r i q

I

using the identity h p0 r i q

h qri C p0 D p 0  1; q

this gives h kpp0 i q

D p0

h kp i q

h q ri C p0  1  p0 ; q

7 Proof by Kronecker [43]

27

which in turn implies h kpp0 i q



h kp i q

C

h k0p0 i q

mod 2:

This formula yields the relation  pp0 

D

q

 p  p 0  : q q

(3.26)

Applying formula (3.23) to this and switching p 0 and q gives  p0  pq

D

 p0  p 0  : p q

(3.27)

The formulas (3.24)–(3.27) show that the symbols . pq / and . pq / occurring in (3.23) satisfy the same relations as the Jacobi symbol. In order to show that they agree we have to show that p D C1 if p is quadratic residue modulo q, and q p  D 1 if p is quadratic nonresidue modulo q. q These relations do hold. In order to prove the first one, put p 0 D p in (3.26); using (3.24) we immediately get .p2 =q/ D C1. As for the second claim, assume that there exists an integer p such that . pq / D 1; then it is obvious by (3.24) and (3.26) that every nonresidue satisfies the same relation. Such an integer p exists. For q D 8n C 1 this was proved by GAUSS in his first proof; thus it remains to prove the claim for q D 8n C 3, 8n C 5, 8n C 7, or equivalently, for q  1 mod 4

and q  5 mod 8:

Assume first that q  1 mod 4; then setting p D 2q  1 in (3.25) shows . pq / D 1. For p  5 mod 8 and p D qC1 the same formula (3.25) gives . pq / D . 2p1 /D 2 p p .p1/=2 D 1, and then . q / D 1 by (3.23). .1/ Now we have shown that . pq / is identical to the Legendre symbol, and now (3.23) is the reciprocity law.

28

3 Proof by Reduction

8 Proof by Bouniakowski [3] 1. If a and r (1 < r < 2a  1) are positive odd and coprime integers, and if p D r1 2anCr is an odd prime number, then the numbers 1; 2; : : : ; p1 2 or 1; 2; : : : ; anC 2 are represented by the system 8 1; ˆ ˆ ˆ ˆ ˆ 2; ˆ ˆ ˆ ˆ ::: ˆ ˆ < r1

1 C a; 2 C a; ::: r1 ; 2 2 C a; rC1 rC1 ˆ ; C a; ˆ 2 2 ˆ ˆ ˆ : : : : : : ˆ ˆ ˆ ˆ a  1; a  1 C a; ˆ ˆ : a; 2a

: : : 1 C .n  1/a; 1 C na; : : : 2 C .n  1/a; 2 C na; ::: ::: ::: r1 : : : r1 C .n  1/a; 2 2 C na; : : : rC1 C .n  1/a; 2 ::: ::: : : : a  1 C .n  1/a; ::: na

(3.28)

Denote the product of all terms in the row with first term r by .r ; a/, that is, put .r ; a/ D r .r C a/.r C 2a/ : : : ; then we get 12

p1 D .1; a/.2; a/    .a; a/: 2

(3.29)

Since the numbers 1; 2; : : : ; a coincide modulo a with the numbers 0, r1 , . . . , r.a1/=2 , . . . , a  r1 , . . . , a  r.a1/=2 , where we have put r  r mod a; then (3.29) can be written in the form 12

Y Y p1 D .0; a/ .r ; a/ .a  r ; a/; 2 

(3.30)



where the product is over  D 1; : : : ; a1 . 2 Now BOUNIAKOWSKI considers the behaviour of the individual factors .A/

.0; a/

.B/ .r ; a/

.C / .a  r ; a/

with respect to the modulus p. (A)

.0; a/ D a  2a  3a   

Each factor of .0; a/ is congruent modulo p to some positive multiple .ka/ of a; p1 and since n < p1 2 , we always have k < 2 .

8 Proof by Bouniakowski [3]

(B)

29

.r ; a/ D r  .r C a/  .r C 2a/   

BOUNIAKOWSKI assumes that r is congruent modulo p to a negative multiple of a, i.e., that r  ka mod p. Substituting this value for r D r  aq we get r  aq  ka mod p

or, since r C 2an D p;

ka  2an  aq mod p

or

k  2n C q mod p: The assumption on r is therefore correct: r is, as the last congruence shows, indeed congruent modulo p to some negative multiple (> n) hof a. i a1 2 r , as can be seen from The maximal value of  is a1 2 , and that of q is a r D r  aq . i h From

a1 r 2

a

D

r1 2

C

ar 2a

we get

a1 r 2

a

Thus the maximum of k is .a  1/n C r  ka mod p n 2, then   0 and also  C 0 are even, and we get fΠpC1  C ΠqC1 g  4 4 0 mod 2. Now assume that p D 4 C 3 and q D 40 C 3; then we get the same result by a similar reasoning. As for a1 .n C n0 /: it follows from p  q D 2a.n  n0 /  0 mod 2 that 2 0 .n C n0 / are even integers. n  n and therefore also n C n0 as well as a1 2 Thus we find  p  q  p1 D .1/ 2 ; where p  q mod 4: q p (II) p q  2 mod 4. Then p D 4C3, q D 40 C1 or p D 4C1, q D 40 C3. Both cases can be reduced to a single one because the condition p > q is not needed here. Assume therefore that p D 4 C 1 and q D 40 C 3. Then we have 

nh p C 1 i 4

C

h q C 1 io 4

  C 0 C 1;

as well as p  q D 2a D 2f2.  0 /  1g, hence   0 D C 1  a1 mod 2. Thus we get  C 0 C 1  aC1 2 2  p  q  q

p

D .1/

p1 a1 0 2 C 2 .nCn C1/

aC1 2

and

:

Taking into consideration that p  q D 2a.n  n0 / D 2a, we find n  n0 D 1 and thus n C n0 C 1  0 mod 2. This shows  p  q  p1 D .1/ 2 ; where p  q  2 mod 4: q p Collecting the cases (I) and (II) we get the well known formula.

9 Proof by Schering [67, 68] We assume GAUSS’s Lemma . pq / D .1/ , and denote by p 2p , q, q

...,

q1 p ; 2 q

then

kp q

kp q

one of the numbers

will contribute to , i.e. will be congruent modulo q to

a negative minimal residue, if there is an integer between

kp q

and

kp q

C 12 . Thus if h

runs through the integers from 1 to , where is an arbitrary integer >

kp q

then the

32

3 Proof by Reduction

number of positive values of the expression

kp q

C 12  h minus the number of positive

 h equals 1 or 0 according as kp is congruent modulo q values of the expression kp q to a negative or a positive minimal residue. As a formula: n kp n kp o o 1 card: pos: C  h  card: pos:  h D 1; 0: hD1 hD1 q 2 q Thus we get for : .q1/=2 n

X



kD1

Since

q1 p 2 q



card: pos:

 kp q

hD1

C

is the maximal value of D

 kp  o 1  h  card: pos:  h mod 2: hD1 2 q

kp , q

we may put

C

p1 1i D D p0 : 2 2

h q1 p 2

q

C 12  h by pC1  h, which is clearly If we also substitute h in the expression kp q 2 allowed since the number of positive terms in a sequence is independent of the order of these terms and since the characteristic property 1  h  is preserved, and if we moreover divide the individual terms by the positive number p, which is also allowed since only the signs matter, then we get 0



q n X kD1

p0

card: pos: hD1

k q

C

k p0 h 1 h o  card: pos: mod 2:  C hD1 p 2 q p

If we now denote the exponent belonging to the residue q and the modulus p by , which implies . pq / D .1/ , then we get in a similar manner 0



p n X kD1

q0

card: pos: hD1

h p

C

h q0 k 1 k o  card: pos: mod 2:  C hD1 q 2 p q

These two congruences for  and  imply the following:  C   card: pos: h;k

k q



h k h C card: pos: mod 2:  h;k p p q

q1 The two double sums each contain p1 terms. Since p and q are primes, the 2 2 k h term q  p does not vanish. Thus either kq  ph or ph  kq is positive, and this shows that

C  which is what we wanted to prove.

p1q1 mod 2; 2 2

10 Proof by Petersen [63, 64]

33

10 Proof by Petersen [63, 64] Let p and q > p be two distinct positive odd primes. For 2n C 1 D 1; 3; 5 . . . , 4q  2, PETERSEN chooses m in such a way that .2n C 1/p  2mq D r;

(3.36)

where r is an odd integer between q and q. Let  denote the number of negative values of r; then clearly .p=q/ D .1/. Among the residues in (3.36), PETERSEN singles out those that lie between Cp and p. As a condition for this he gets the equation .2n0 C 1/q  2m0 p D r, or, by adding and subtracting pq in (3.36): .p  2m/q  .q  2m  1/p D r:

(3.37)

This implies that r is between Cp and p in (3.36) for p  2m D 1; 3; : : : ; p  2; that is, for m D 1; 2; : : : ; p1 . 2 Now replace  by  and switch the roles of p and q; then .q=p/ D .1/ . Moreover it is seen that  can be derived from (3.37) in the same way as  from (3.36). Thus .p=q/ and .q=p/ will have the same or the opposite sign according as the number of residues r between p and q is even or odd. For such residues q < .2n C 1/p  2mq < p we get by putting m D n  k and p D q  2a: 2m C 1
jp2 j > jp3 j > : : :.7 By Assumption (3.39) we conclude that the relation .x; y/ holds for ˙1, pC1 , hence by (3.41) also for p and pC1 , finally by (3.42) for p and p1 etc.; thus the relation holds for p and p1 as well as for p1 and p: in fact, if the initial values x D ˙1, y D pC1 show the validity of the relation for p and p1 , then Assumption (3.40) would show that it also holds for p1 and p. This theorem can also be formulated like this: Every relation .p; q/ between two arbitrary odd coprime integers p and q is valid in general if it holds in the following cases: 1/ ˙ 1; qI

2/ p; ˙1I

4/ p; q C 20 q;

3/ p C 2q; qI

where  and 0 are integers, always under the assumption that the relation holds for .p; q/. Thus the relation holds in general if the relations I / .˙1; q/I

II/ .p; ˙1/I

IV/ .p; q C 20 q/;

III/ .p C 2q; q/I

hold whenever .p; q/ is true.

The quadratic reciprocity law in its simplest form can be stated as follows:  p  q  q

p

D .1/

p1 q1 2 2

;

where p and q are positive odd primes. In order to show that this formula is valid in general, it must be shown, taking its symmetry into account, that  "  q  q  p C 2q  q

7

"

D .1/

"1 q1 2 2

;

where " D ˙1;

 pC2q1 q1 q 2 ; D .1/ 2 p C 2q

According to Kronecker and Weierstrass, jxj denotes the absolute value of x.

(3.43) (3.44)

12 Proof by Busche [4]

37

assuming that  p  q  q

p

D .1/

p1 q1 2 2

:

(3.45)

"1 q1

2. Since . q" / D .1/ 2 2 and . q" / D C1, the correctness of (3.43) is immediately clear. q In order to verify (3.44), BUSCHE exploits a relation between . pC2q / and q . p /, namely the relation between GAUSS’s characteristic numbers attached to these symbols. Let us write q  p

D .1/ ;

(3.46)

then kq (k D 1; 2; : : : ; p1 ) will contribute to  if 2 (

kq D hp C

pC1 2

Cr

kq D hp C p  r

or

(3.47)

0

where r and r 0 are positive integers less than p2 . Now we look for a relation between , the maximal value attained  and h. Since the maximal value attained by k is p1 2 , where we have to remark that h does not necessarily attain q3 . As h by h is q3 2 2 q3 runs through the integers from 1 to 2 , let h denote the number of values kq for each h; thus h may also be defined as the number of solutions k of kq D hp C

pC1 C r; 2

r
0 for every h; if q > p, on the other hand, then h D 0 or 1. This can be seen by comparing the maximal values attained by h and k. If moreover q P

X

.q3/=2

D .1/M ;

M D

hD0

Mh

(3.49)

38

3 Proof by Reduction

for P D p C 2q, then Mh is the number of solutions of ( C r or Kq D hP C P C1 2 Kq

D hP C P  r 0

(3.50)

with r; r 0 < p2 . Now assume first that h D C1, i.e., that Eqs. (3.47) are solvable; this implies that, for positive , we have fk C .2h C 1/gq D hP C

P C1 C r; 2

fk C .2h C 2/gq D hP C P  r 0 : Setting K1 D k C .2h C 1/;

K2 D k C .2h C 2/

(3.51)

gives (

P C1 2

K1 q

D hP C

C r;

K2 q

D hP C P  r 0 :

(3.52)

This implies that (3.50) has K2  K1 C 1 pairwise distinct integral roots, hence that Mh D K2  K1 C 1 D  C 1 D  C h :

(3.53)

In order to show that the Mh values of K modulo P are distinct as well as different from those that occur for other values of h it is sufficient to remark that q < P . Next, if h D 0, then the interval from hp C pC1 to hp C p does not contain a 2 to hP C p C p multiple of q, hence the same is true for the interval from hP C P C1 2 p1  hp C , . . . , hP C p C p  hp C p mod q. Now at least  because hP C P C1 2 2 P C1 of the numbers hP C 2 , . . . , hP C P are divisible by q, because their cardinality C q C 1 > q; on the other hand there are at most  multiples of q since is p1 2 integers are not divisible by q. Thus the equation the first pC1 2 Kq D hP C

P C1 Cr 2

has  roots; we have Mh D  D  C h ;

(3.54)

hence we have (under the assumption that q > p) in general Mh D  C h :

(3.55)

12 Proof by Busche [4]

39

Above we have seen that every h yields a nonzero Mh , hence we get M D

X q1 C h ; 2

i.e., M D

  q q1 q q1 D .1/ 2 :  C  or 2 p C 2q p

Since q is odd, we also have 

 q q1 q D .1/ 2 q : p C 2q p

Now  p C 2q  q

D

p  q

;

hence  p C 2q  q

  p  q  q1 q D .1/ 2 q p C 2q q p

which is what we wanted to prove.

D .1/

q1 p1 q1 2 qC 2  2

D .1/

q1 pC2q1 2  2

;

(3.56)

Chapter 4

Eisenstein’s Proof Using Complex Analysis

1. Let p and q be two distinct positive odd primes and r the positive minimal residues modulo q; then we will have pr  r 0 or pr  r 0 mod q, where r 0 again denotes the positive minimal residues modulo q.1 Thus we have pr r0 D Cf q q

or

D

r0 C f; q

or

 sin

where f and f 0 are integers. This implies  2r  2r 0

D sin sin p q q

2r 0

: q

The properties of the sine function expressed by the above equation immediately leads to the result 0

pr 

sin 2r qp

mod q;

0

sin 2rq

and this in turn shows that p

1

q1 2

Y

r

Y

0

r

0

Y sin 2r qp

sin

2r 0

q

mod q;

Eisenstein’s proof was published in [16].

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_4

41

42

4 Eisenstein’s Proof Using Complex Analysis

where the product is over all positive minimal residues modulo q. Since the r 0 are, up to their order, equal to the r, we find p  q

q1

D

2 sin 2rp

Y q

rD1

sin

2r

q

and similarly

q p

p1

D

2 Y sin 2p

q

D1

sin 2

q

:

(4.1)

2. According to the above, E ISENSTEIN has to deal with expressions of the form sin tv , where t is an odd prime.2 Assume now that the expression sin.tsin2/v is a sin v v polynomial in sin v; then it is immediately clear that it is an even function of sin v, 2/v and that cos.t has the same property. Now from cos v sin tv D sin.t  2/v  cos 2v C cos.t  2/v  sin 2v sin v tv is also an even polynomial in sin v whose degree is two units larger we see that sin sin v 3v than the degree of sin.tsin2/v . Thus, recalling that sin D 3  4 sin2 v, we find in v sin v general:

sin tv D at 1 sint 1 v C at 3 sint 3 v C : : : : sin v EISENSTEIN now transforms the right-hand side of the preceding equation into a tv product. To this end we have to know the roots of sin D 0 as well as the coefficient sin v at 1 . Assuming sin.t  2/v t 3 D .1/ 2 2t 3 sint 3 v C : : : sin v we find by a simple calculation sin tv t 1 D .1/ 2 2t 1 sint 1 v C : : : : sin v From sin 3v 31 D 3  4 sin2 v D .1/ 2 231 sin31 v C : : : sin v we now find that this formula for

2

sin tv sin v

is valid in general.

For deriving the subsequent theorems concerning

sin tv sin v

it is sufficient to assume that t be odd.

4 Eisenstein’s Proof Using Complex Analysis

Thus if we denote the then we have3

p1 2

43

roots of the equation

sin tv sin v

D 0 by D 1 , . . . , .p1/=2 ,

Y sin tv t 1 D .1/ 2 2t 1 .sin2 v  2 /: sin v 3. According to the results just developed we have, if we denote the equation P D

2rp

q sin 2r

q

sin

D 0 by , the

q1 2

roots of the equation Q D

p1 roots of 2 2q

sin p sin

2

p

the

D 0 by

, and the variable sin v by x: P D .1/

p1 2

2p1

Y

.x 2   2 /;

Q D .1/

Introducing the abbreviations ˛ D sin 2

and ˇ D p

q1 2

2q1

2r

, q

Y .x 2  2 /:

then ˛ and ˇ attain

p1 2

and q1 distinct values, respectively; at the same time, they satisfy the equations 2 P D 0 and Q D 0, respectively. Thus we find P D .1/

p1 2

2p1

Y .x 2  ˛2 /;

Q D .1/

q1 2

2q1

Y

.x 2  ˇ 2 /:

Now in P we have x D ˇ, and in Q we have x D ˛; this shows that P D .1/

p1 2

2p1

Y .ˇ2  ˛ 2 /;

Q D .1/

q1 2

2q1

Y .˛2  ˇ 2 /:

Using (4.1) this implies (p  qq  p

D D

Q

2 ˛;ˇ .ˇ Q 2 ˛;ˇ .˛

˚  p1 .1/ 2 2p1 ;  Q ˚ q1  ˇ 2 / ˛;ˇ .1/ 2 2q1 ;  ˛2 /

Q

˛;ˇ

hence (p  qq  p

D .1/

p1 q1 2  2

D .1/

p1 q1 2  2

2

.p1/.q1/ 2

2

.p1/.q1/ 2

Q

2 ˛;ˇ .ˇ 2 ˛;ˇ .˛

Q

 ˛ 2 /;  ˇ 2 /;

and this implies Q 2 Y .˛2  ˇ 2 / .ˇ  ˛ 2 / : D Q 2 D p ˛2  ˇ2 .˛  ˇ 2 /

 p  q  q

3

In Eisenstein’s article, the power of 2 erroneously occurs in the denominator.

44

4 Eisenstein’s Proof Using Complex Analysis

since ˛ ¤ ˇ as p and q are primes, the quotient finally gives  p  q  q

p

D .1/

˛ 2 ˇ 2 ˛ 2 ˇ 2

p1 q1 2  2

is always equal to 1, and this

:

Chapter 5

Proofs Using Results from Cyclotomy

1 Proof by Gauss (7th Proof) [34] and Lebesgue (2nd Proof) [52] 1. Let  be a primitive root of the equation x x11 D 0, where p is a positive odd prime, and let g be a primitive root modulo p; then we can order the roots of x p1 1 D 0 in the following way: x1 p1

2

4

; g ; g ; : : : ; g

p3

3

and g ; g ; : : : ; g

p2

:

The expressions 3

y1 D  g C  g C : : : C  g

p2

;

2

y2 D  C g C : : : C g

are called quadratic1 periods of the cyclotomic equation property

x p1 1 x1

p3

D 0. Using their

y1  y2 D .1  /.3  3 /    .pC2  p2 / and the relation .x  2 /.x  4 /    .x  2.p1/ / D x p1 C x p2 C : : : C 1 we find .y1  y2 /2 D .1/

1

[FL] Baumgart uses the expression

p1 -termed 2

p1 2

p:

periods.

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_5

45

46

5 Proofs Using Results from Cyclotomy

Now y1 C y2 D 1, hence we get y1 y2 D

1  .1/ 4

p1 2

p

:

Thus the two periods y1 and y2 are roots of the quadratic equation f .x/ D x 2 C p1 x C 14 .1  .1/ 2 p/ D 0. 2. GAUSS and LEBESGUE now study under which conditions the congruence f .x/  0 mod q;

(5.1)

where q as well as p is an odd positive prime, has real integral roots. This condition can be expressed in two different ways, and a comparison will then yield the quadratic reciprocity law. The congruence (5.1) turns, after substituting y D 2x C 1;

(5.2)

into the congruence y 2  .1/

p1 2

p mod q:

(5.3)

Thus if the congruence (5.1) has real roots, then so does (5.3); conversely, if (5.3) is solvable, then the substitution (5.2) shows that this is also true for (5.1). This shows that the congruence (5.1) is solvable if  .1/ p1  2 p q

D C1;

or if .1/

p1 q1 2 2

p

q1 2

 1 mod q:

(5.4)

On the other hand, f .x/  0 mod q does not have real integral roots if .1/

p1 q1 2 2

p

q1 2

 1 mod q:

Moreover, the identity x q1  1  .x  1/.x  2/    .x  q C 1/ mod q x q  x  x.x  1/.x  2/    .x  q C 1/ mod q

(5.5)

1 Proof by Gauss (7th Proof) [34]Gauss and Lebesgue (2nd Proof) [52]

47

can be transformed into .y  yh /q  .y  yh /  .y  yh /.y  1  yh /.y  2  yh /    .y  q C 1  yh / mod q h

by the substitution x D y  yh (h D 1; 2). Now yh D x g C x g so if we write

2Ch

C : : : C xg

q  gk mod p

(5.6) p3Ch

,

(5.7)

we find yhq  yhCk mod q or .y  yh /q  y  yhCk mod q or finally .y  yh /q  .y  yh /  yh  yhCk mod q. Thus congruence (5.6) implies .y  yh /.y  1  yh /.y  2  yh /    .y  q C 1  yh /  yh  yhCk mod q; which in turn immediately implies the following formula: .y  y1 /.y  y2 /.y  1  y1 /.y  1  y2 /    .y  q C 1  y1 /.y  q C 1  y2 /  .y1  y1Ck /.y2  y2Ck / mod q Since f .y/ D .y  y1 /.y  y2 /, the last congruence can be written in the form f .y/  f .y  1/    f .y  q C 1/  .y1  y1Ck /.y2  y2Ck / mod q:

(5.8)

If f .y/  0 mod q only had real integral roots, then f .y/; : : : ; f .y  q C 1/ mod q would represent a complete residue system modulo q, and we would get f .y/  f .y  1/    f .y  q C 1/  0 mod q: Conversely, if this condition is satisfied, then the congruence f .y/  0 mod q and hence f .x/  0 mod q would be solvable in real integers. Taking the congruence (5.8) into account we can say: f .x/ has q real roots modulo q if  D .y1  y1Ck /.y2  y2Ck /  0 mod q, and is not solvable modulo q if  D .y1  y1Ck /.y2  y2Ck / 6 0 mod q. It is clear that this depends on the value of k, which was defined by q  gk mod p. Thus if k  0 mod 2, i.e., . pq / D C1, then yh D yhCk ; if k  1 mod 2, on the other hand, then yhC1 D yhCk . This shows: f .x/  0 mod q has real integral roots if . pq / D C1, and does not have real integral roots if . pq / D 1. Comparing this result with the one expressed by our formulas (5.4) and (5.5) the quadratic reciprocity law follows.

48

5 Proofs Using Results from Cyclotomy

2 Gauss’ Fourth Proof [27] 1. Let p and q denote as usual two distinct odd primes, let  D e 2 i=q , and let a and b denote the quadratic residues and nonresidues modulo q; then ( Pq1

2 p D D1  P ap P a C b

G. piq / D 1 C 2

P a

ap ;

(5.9)

bp D p C 2p C : : : D 1;

hence2 G

 pi  q

D

 p  X  p  p  X  X  p D p D  : q q q q q 





The last equation can also be written in the following form: G

 pi  q

D

p   i  p  2 G D .1 C  C 4 C : : : C .q1/ / : q q q

(5.10)

2. GAUSS first determines3 the value of G. qi /. Using the system of equations 8 q1 1 ˆ ˆ 1 ˆ ˆ ˆ < 1q2 12

ˆ ::: ˆ ˆ ˆ ˆ : 1q.q1/ 1q1

D

11 1 12 12

D

1.q1/ 1q1

D

D 1 ; D 2 ;

::: D .q1/

he forms the series 8 1q1 .1q1 /.1q2 / ˆ ˆ  C 1, and if we observe that 1  m 1  m1 m1 .1  C1 / D C ; C1 C1 1 1 1  C1 then we get .q  1;  C 1/ D m2 .q  2; / C .q  2;  C 1/: Applying this equation to (5.11) we find f .; q  1/ D .1  q2 /  .1  q3 /.q  2; 1/ C .1  

q4

/.q  2; 2/  .1  

q5

(5.12) /.q  2; 3/ ˙ : : :

Now .1  q1.C1/ /.q  2; / D .1  q2 /.q  3; /; hence f .; q  1/ D .1  q2 /f1  .q  3; 1/ C .q  3; 2/  : : :g or f .; q  1/ D .1  q2 /f .; q  3/:

(5.13)

Since q  1 mod 2 we find f .; q  1/ D .1  q2 /f .; q  3/; f .; q  3/ D .1  q4 /f .; q  5/; :::

:::

f .; 2/ D .1  /; and multiplying these equations yields f .; q  1/ D .1  /.1  3 /.1  5 /    .1  q2 /:

(5.14)

Thus we have two developments for f .; q 1/. Combining these two results shows that 1 C 1 C 3 C : : : C q.q1/=2 D .1  /.1  3 /    .1  q2 /:

50

5 Proofs Using Results from Cyclotomy

Taking into account that .q2 / D 2 for integral  we find 1 C 2 C 6 C 12 C : : : C q.q1/ D .1  2 /.1  6 /    .1  2.q2/ /: Multiplying both sides by . .

q1 2 2 /

q1 2 2 /

C 2C.

D   3    q2 gives

q1 2 2 /

C : : : C q.q1/C.

q1 2 2 /

D .  1 /.3  3 /    .q2  .q2/ /: Thus we have found G

i  q

D .  1 /.3  3 /    .q2  qC2 /;

(5.15)

or, using the fact that    D .q  qC /: G

i  q

D .1/

q1 2

.2  2 /.4  4 /    .q1  qC1 /:

(5.16)

Multiplying (5.15) and (5.16) we get G2

i  q

D .1/

q1 2

q

q1 2

.1  2 /.1  4 /    .1  2.q1/ /;

or, since  is a primitive root of x q  1, G2

i  q

D .1/

q1 2

q:

This implies G

i  q

D ˙i .

q1 2 2 /

p

q

and G

 pi  q

D ˙i .

q1 2 2 /

 p p q

q:

(5.17)

, we In order to find the sign of G we return to (5.15). Since    D 2i sin 2  q find G

i  q

D .2i /

q1 2

sin

6

10

.q  2/2

2

 sin  sin    sin : q q q q

2 Gauss’ Fourth Proof [27]

51

The values 2

, . . . .q2/2

are all smaller than 2 ; now recall that q is odd and q q distinguish the following two cases: 1. q D 4n C 1. Then

q1 4

of the angles are greater than , hence G

i  q

Di

q1 2

q1 4

.1/

C D C;

where C denotes a positive constant. 2. q D 4n C 3. In this case the number of angles greater than is G

i  q

Di

q1 2

.1/

q3 4

q3 4 ,

and we find

C D iC:

Combining these two cases finally shows G

i  q

D i.

q1 2 2 /

p

q

and G

 pi  q

D i.

q1 2 2 /

 p p q

q:

(5.18)

3. Since we also assumed that p be odd we now find G

 pi   qi   p  q  p1 2 q1 2 p p G D i . 2 / i . 2 / p q: q p q p

(5.19)

Now we have, according to the definition of Gauss sums, G

q1 p1  2  pi   qi  X X  p C 2 q 2 i X X .pCq/2 2 i p e q D e pq G D q p  D1 D1



since 2 q .p C q/2 2 p C D  2: q p pq It is clear that p C q attains pq different values modulo pq and forms, as can be seen from p.  0 / D q.  0 /, a complete set of residues modulo pq. Thus we get G

 pi   qi   i  pq1 2 p G DG D i . 2 / pq; q p pq

and observing (5.19) now yields i.

pq1 2 2 /

 p  q  p1 2 q1 2 p p p i . 2 / C. 2 / p q: pq D q p

52

5 Proofs Using Results from Cyclotomy

Since the square roots are taken to be positive, this shows that i.

pq1 2 2 /

 p  q  p1 2 q1 2 i . 2 / C. 2 / ; q p

D

(5.20)

which immediately implies the quadratic reciprocity law.

3 Gauss’ Sixth Proof [29] 1. Assume that p and q have their usual meaning and let G denote the series 2

G D x  xg C xg  : : :  xg

p2

;

(5.21)

where g is a primitive root modulo p. Then it follows from properties of binomial 2 p2 coefficients that G q  .x  x g C x g  : : :  x g /q  0 mod q, or, since q is odd, that G q  Gq  0 mod q;

2

where Gq D x q  x qg C x qg  : : :  x qg

p2

:

(5.22)

If moreover q  g mod p, then the system of equations q D g  C f1 p;

qg D gC1 C f2 p;

:::;

qgp2 D gCp2 C f3 p

implies 

x qg  x qg

C

D .1  x p /f .x/;

(5.23)

where f .x/ is a polynomial in x. Thus we find ˚  C1 Cp2  D .1  x p /W; Gq  x g  x g ˙ : : : ˙ xg

(5.24)

where W is also a polynomial in x. The exponents of the p  1 terms inside the brackets are just the integers 1; 2; : : : ; p  1 since g is a primitive root modulo  C1 ˙ : : : D ˙G. The sign of p. Since the signs alternate, we see that x g  x g G is that of .1/p x, and since p is odd we conclude that ˙G D .1/ G. p1 p1 From q  g mod p we then find q 2  .g 2 /  . pq / mod p, and since g

p1 2

 1 mod p, this implies .1/ D

q  p

3 Gauss’ Sixth Proof [29]

53

and Gq 

q G D .1  x p /W: p

(5.25)

2. Now consider the system of identities 2 C1

C : : : C xg

p2 C1

D 0;

3 Cg

C : : : C xg

p1 Cg

D x gC1 .x g

CxG  x 2 C x gC1  x g x g G  x 2g C x g x

g p2

Gx

2g p2

2 Cg

Cx

 xg

g p1 Cg p2

C ::: C x x

g p2 C1

fx

g 2p4 Cg p2

D

g p1 1

 .x g

1

p 1

p2

 1/;

 1/  : : :g:

Adding these equations gives ˝ D G 2  f .x g

0 C1

/ C f .x gC1 /  : : : C f .x g

p2 C1

/;

(5.26)

where ˝ denotes the sum of the expressions on the right-hand side of the above p2 system of equations and where we have set f .x  / D 1 C x  C x g C : : : C x g . p It is easily seen that ˝ is divisible by 1  x p , hence by 1x ; on the other hand 1x f .x  / is, because g is a primitive root modulo p, divisible by 1  x p , hence by p 1x p . Thus f .x  / will be divisible by 1x if 1x 1x 1  xp 1  x p  0 mod : 1x 1x For a proof we have to distinguish two cases. (I)  a n d p a r e c o p r i m e. Then y D hp C 1 for integers y and h, hence 1  x p 1  x p 1  x p 1  x hp 1  x p 1  x y  x  ; W D  1x 1x 1x 1  x 1  x 1  xp p

and this implies that f .x  / is divisible by 1x . 1x (II)  a n d p a r e n o t c o p r i m e. Then 2

f .x  /  p D x  f.x g  1/ C .x g  1/ C : : : C .x g

p2

 1/g;

p

. and this immediately implies that f .x  /  p is divisible by 1x 1x Collecting everything and recalling that g0 C 1, g C 1, . . . , g p2 C 1 represent the numbers 2, 3, . . . , p in some order, we can deduce from (5.26) ˝ D G 2  .1/

p1 2

 p1  1  xp f x g 2 C1  0 mod 1x

(5.27)

54

5 Proofs Using Results from Cyclotomy

or, if Z denotes a polynomial in x, G 2  .1/

p1 2

pD

1  xp Z: 1x

(5.28)

From (5.28) we immediately deduce G q1  .1/

p1 q1 2 2

p

q1 2

D

1  xp Y: 1x

(5.29)

3. Using Eqs. (5.23), (5.24), (5.28) and (5.29), the reciprocity law can be proved easily. First we observe that (5.23) and (5.24) imply n q o qGX D G qC1  G .1  x p /W C G ; p where X denotes a polynomial in x defined by (5.22) as G q  Gq D qX: Moreover, from (5.29) we get q n p1 q1 q1 1  xp o 2 qGX D .1/ 2 2 p 2 C G2; Y G  G.1  x p /W  1x p or, using (5.28), n  q o p1 q1 q1 p .1/ 2 2 p 2  p  q 

o p1 p1 1  xp n C YG2  WG.1  x/ : Z .1/ 2 q 2  C 1x p

qGX D .1/

p1 2

(5.30)

p

U C T , where U According to (5.21), G has degree p  1. If we put GX D 1x 1x and T are polynomials in x, then T will be a polynomial of degree less than p  1. Plugging the last equation into (5.30) we get qT  .1/

p1 2

n  q o p1 q1 q1 p .1/ 2 2 p 2  p  q i pn h q1 p1 q1 1x D Z .1/ 2 2 p 2  1x p o C YG2  WG.1  x/  qU ;

(5.31)

where the degree of the left-hand side is less than p  1. Now Z; Y; W are polynomials in x, hence the degree of the right-hand side is bigger than p  1.

4 Proof by Cauchy [8],Cauchy Jacobi [56, p. 391], Jacobi Eisenstein [15]

55

Thus the equation above can hold only if both sides vanish. Thus we find qT D .1/

p1 2

n  q o p1 q1 q1 p .1/ 2 2 p 2  p

or .1/

p1 q1 2 2

p

q1 2



q p

 0 mod q;

and this is what we wanted to prove.

4 Proof by Cauchy [8], Jacobi [56, p. 391], Eisenstein [15] GAUSS has shown that G

 qi  p

D

q  i  G p p

and G 2

i  p

D .1/

p1 2

p:

(5.32)

This implies without problems that G qC1

i  p

G

q   i   i   qi  p1 qC1 qC1 G D .1/ 2 2 p 2  G2 p p p p

or G

 i h p

Gq

i  p

G

 qi i p

D .1/

p1 2

n  q o p1 q1 q1 p .1/ 2 2 p 2  : p

(5.33)

Now G

i  p

D

p1   X    ; p D1

G

 qi  p

D

p1   X  q  ; p D1

where  denotes a primitive root of x p D 1. Thus we have Gq

i  p

DG

 qi  p

C q.A0 C B 0  C C 0 2 C : : :/;

where A0 , B 0 , . . . are integers. Now we set  q i h p1 q1 q1 p1 .1/ 2 p X D .1/ 2 2 p 2 p

(5.34)

56

5 Proofs Using Results from Cyclotomy

and get X DG

i  p

 qŒA0 C B 0  C : : :

D qŒA C B C : : :;

or

where A, B, . . . again denote integers. Plugging in 2 , . . . p1 for  and adding the resulting equations gives .p  1/X D qŒ.p  1/A  B  C  : : ::

(5.35)

Since q is a prime and we can assume without loss of generality that p  1 < q, (5.34) and (5.35) now imply .1/

p1 q1 2 2

p

q1 2



q  p

 0 mod q:

This is our well-known formula.

5 Second Proof by Eisenstein [14] 1. Let p P be an odd prime and let r run through a complete system of residues modulo p; then r . pr / D 0, and similarly ./ D

n X  r o r

p

D 0:

(5.36)

Multiplying out immediately gives ./ D

X  ˛1   ˛   ; p p ˛ ;:::;˛ 1

(5.37)



where the summation for each ˛i is from 1 to p  1. Introducing the sum4 .; k/ D

X P

˛ 

˛i k mod p

1

p



˛   : p

we find ./ D

4

.; 0/ C

.; 1/ C : : : C

.; p  1/ D 0:

[FL] Such sums are nowadays called “multiple Jacobi sums”.

(5.38)

5 Second Proof by Eisenstein [14]

57

Setting ˛1  ˇ1 , ˛2  kˇ2 , . . . , ˛  kˇ mod p, then 1, and therefore .; k/ D Now if  is even, then

.; k/ D

.; 1/ D

 k  p

P

˛i D k

P

.; 1/:

ˇi ,

P

ˇi D

(5.39)

.; 1/, or

.; 2/ D : : : D

.; p  1/;

(5.40)

hence using (5.38) .; 0/ C .p  1/ .; 1/ D 0: If  is odd, on the other hand, then .; 1/ C : : : C

(5.41)

.; k/ D . pk / .; 1/, which implies that

.; p  1/ D

.; 1/

X r  p

D 0;

hence .; 0/ D 0:

(5.42)

P The defining equation .; / D . ˛p1 /    . ˛p /, where the summation is over all P ˛i with ˛i   mod p, can also be written in the form .; / D

X n ˛  X  ˛1  p

p



˛

1

p

o ;

from which we see that .; / D

X n ˛ 

o .  1;   ˛ / I

p

in the special case  D 0 this shows that .; 0/ D

X  ˛  p

.  1; ˛ /;

or, using (5.39): .  1; ˛ / D

 ˛ 1 

p

.  1; 1/:

58

5 Proofs Using Results from Cyclotomy

This implies .; 0/ D

X  ˛   1  p

p

.  1; 1/:

For even  this shows that .; 0/ D

 1  p

.  1; 1/  .p  1/;

or, using (5.41):  1  .  1; 1/; .; k/ D  p

  0 mod 2:

(5.43)

For odd  the recursion formula gives us .; k/ D D

X  ˛  p k  p

.  1; k  ˛ /

.  1; 0/ C

X  ˛  p

.  1; k  ˛ /:

Now formula (5.40) implies .; k/ D

k 

.  1; 0/ C

p k  f .  1; 0/  D p

.  1; 1/

X  ˛  p

.  1; 1/g;

or, using (5.41):  1  .; k/ D  p .  1; 1/; p

  1 mod 2:

(5.44)

From Eqs. (5.43) and (5.44) we easily deduce, for integers , the following system of equations: .2 C 1; 1/ D p  .2; 1/;  1   .2  1; 1/; .2; 1/ D  p :::  1   .2; 1/ D  p

.1; 1/;

5 Second Proof by Eisenstein [14]

59

and multiplying them all together gives .2 C 1; 1/ D .1/2 or, since

 1  p

p  .1; 1/

.1; 1/ D 1, .2 C 1; 1/ D .1/

p1 2 

p :

(5.45)

2. Let q D 2 C 1 be, as p, an odd positive prime. According to the formula just found we have .q; 1/ D .1/

p1 q1 2 2

p

q1 2

:

(5.46)

On the other hand, the defining equation gives .q; 1/ D

˛ 

X P

1

˛i 1 mod p

q



˛  q I p

for ˛1 D : : : D ˛q D ˛ we get q˛  1 mod p. Thus in the sum for exactly one term in which all the ˛i coincide. Now (5.46) implies .q; 1/ D

 ˛ q p

C D .1/

p1 q1 2 2

p

q1 2

.q; 1/ there is

:

P ˛ In D . ˛q1 /    . pq /, the summation is over all ˛i except ˛1 D : : : D ˛q D ˛. From q˛  1 mod p we deduce that 1D

 q  ˛  ; p p

and since q is odd, we find .1/

p1 q1 2 2

p

q1 2



q p

D :

(5.47)

Writing down the sum explicitly we get 8  ˛1  ˛2   ˛q  9  > ˆ ˆ > ˆ  p  p   p  > X < ˛2 ˛3    ˛1 = p p p :

D ˆ     > > ˆ > ˆ      ˛ ; : ˛q ˛1    q1 p p p

(5.48)

60

5 Proofs Using Results from Cyclotomy

Thus can be decomposed into groups in such a way that each group consists of q equal summands. This shows that  0 mod q, and now (5.47) gives .1/

p1 q1 2 2

p

q1 2



q p

mod q;

which is what we wanted to prove.

6 Proof by Liouville [61] Let p be a positive odd prime and  a primitive root of x p D 1. Then xp  1 D .x  2 /.x  4 /    .x  2.p1/ / D 1 C x C x 2 C : : : C x p1 ; x1

(5.49)

and plugging in x D 1 yields p D .1/ Raising this equation to the

p1 2

 p1 p1 2 .  1 /2     2   2 :

q1 th 2

power we get

p  Y ˛q  ˛q mod q;  ˛  ˛ q

.p1/=2

p

q1 2

D .1/

p1 q1 2  2

(5.50)

˛D1

where q denotes a positive odd prime distinct from p. The individual factors of Q.p1/=2 ˛q ˛q are positive or negative according as ˛q is congruent to a positive ˛D1 ˛ ˛ or negative minimal residue modulo p. Applying GAUSS’s Lemma to (5.50) then gives p q

 .1/

p1 q1 2  2

q p

;

a well-known equation.

7 First Proof by Lebesgue [50] 1. Let p and q be distinct positive odd primes and consider the congruence x12 C x22 C : : : C xq2  a mod p:

(5.51)

7 First Proof by Lebesgue [50]

61

Then, as x1 ; : : : ; xq run through the values 1; 2, . . . , p  1, which can happen in .p  1/q

(5.52)

different ways, a will attain certain values. Assume that a vanishes n0q times, that it attains a quadratic residue naq and a quadratic nonresidue nbq times (here and below a denotes a quadratic residue, b a quadratic nonresidue). Then we have5 n0q C naq C nbq D .p  1/q :

(5.53)

Consider a special value a1 of a, and set naq1 D nq . Since all residues can be written in the form ay2 , we find that we have naq D n0q for an arbitrary value of a. The same thing holds for the nonresidues, and thus we get, setting nbq D n0q : n0q C

p1 .nq C n0q / D .p  1/q : 2

(5.54)

Now L EBESGUE computes n0q , nq and n0q . From the well-known formula GD

p1   X  D1

p

q  D

.1/

p1 2

p;

p D 1

due to G AUSS and the formula 1 C  C 2 C : : : C p1 D 0 we get GD

X



2

q D

.1/

p1 2

q p

or G1 D C C: : : D 4

.1/

p1 2

p:

(5.55)



Raising this equation to the qth power we get .G  1/q D n0q C nq

X

a C n0q

X

b :

a

Using

P a

a 

P

b D G and 1 C

P a

a C

P

b D 0 we find

2n0q  nq  n0q C .nq  n0q /G D 2.G  1/q :

(5.56)

If the x run through 0, 1, . . . , p  1, and if we denote the corresponding values n by N , then Nq0 C Nqa C Nqb D p q . This formula can also be used.

5

62

5 Proofs Using Results from Cyclotomy

Since q was assumed to be odd we have .G  1/q D PG  Q and thus 2n0q  nq  n0q D 2Q; where P  .1/ and (5.54) we find6

.p1/.q1/ 4

nq 

p

p1 2

nq  n0q D 2P;

(5.57)

mod q and Q  1 mod q. Comparing (5.57)

p1 q1 p1 .p  1/q C 1 C .1/ 2  2 p 2 mod q: p

(5.58)

2. The congruence x 2  a mod p has, if it is solvable at all, exactly 2 distinct solutions modulo p. This shows nq D 2q S;

where Sq is an integer:

(5.59)

If moreover x12 C x22 C : : : C xq2  a mod p

(5.60)

/ D 1. Conversely, if holds for x1 D x2 D : : : D xq , then qx12  a mod p or . aq p aq 2 . p / D 1, then we can put qx1  a mod p, and (5.60) holds for x1 D x2 D : : : D xq . Now we observe that the number of solutions of (5.60) in which the x are not equal is a multiple of q because q is a prime; thus we find ( Sq D

qR C 1

if . pq / D C1;

qR

if . pq / D 1:

(5.61)

On the other hand we have nq D 2q Sq , hence ( nq 

2q  1 C 1 mod q

if . pq / D C1;

0  1  1 mod q

if . pq / D 1;

and this implies nq D 1 C

q : p

(5.62)

Comparing this with (5.58) and applying Fermat’s Theorem immediately implies the reciprocity law.

6

We get similar formulas for n0q , n0q , Nq0 , Nq , Nq0 .

Chapter 6

Proofs Based on the Theory of Quadratic Forms

1 Preliminary Remarks Recall that the complex of all equivalent forms of the same discriminant is called a form class. If the integers a; b; c in the form1 .a; b; c/ are coprime, then the form is called primitive. If the [greatest common] divisor  of a, 2b, c is 1, then .a; b; c/ is called a form of the first kind, and if  D 2 it is called a form of the second kind. An ambiguous form2 is a form in which the double middle coefficient 2b is divisible by the first. The form .1; 0; D/ is called the principal form of discriminant D; its class is called the principal class. If the outer coefficients of a form are positive, then the form itself is called positive. If z and z0 are representable by the same quadratic form, that is, if z D a˛2 C 2b˛ˇ C cˇ 2 ; z0 D a 2 C 2b ı C cı 2 ; then x 2  zz0 D Dy2 ; thus if z and z0 are coprime to D, then  zz0  p

D C1;

i.e.,

z p

D

 z0  : p

(6.1)

Now we assume that D D pq D 4nC1, where p and q are primes. Then . pz / and have well-defined values (characters), which can be grouped in different ways. If D is a product of two primes, then the different groups are

. qz /

C 1; C1I

C1; 1I

1; C1I

1; 1:

(6.2)

1

This is Gauss’s notation: the symbol .a; b; c/ denotes the binary quadratic form ax2 C 2bxy C cy2 .

2

[FL] BAUMGART uses G AUSS’s Latin word forma anceps.

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_6

63

64

6 Proofs Based on the Theory of Quadratic Forms

For the sake of completeness we remark that if the number of factors of D is , then there are 2 different ways of choosing signs. If the characters . pz / and . qz / attain, for a given determinant D D pq D 4n C 1, the values c1 and c2 , respectively, then the set of all forms of equal determinant and kind with the same characters is called a genus. From (6.1) it follows that every genus consists of a certain number of form classes. The genus containing the principal form and therefore the principal class is called the principal genus. It is immediately clear that the characters of the principal genus for D D pq D 4n C 1 is 1; 1 because . p1 / D 1 D . q1 /.

1.1 Gauss’s Second Proof [25, art. 257], [12, Suppl. IV, X] This proof is based on the following lemma: consider quadratic forms of a given determinant. Then the number of actually existing genera is half as big as the number of possible genera, i.e., half as big as the number of existing characters. The proof of this theorem will not be given here, since in order to do so we would have to reproduce a large part of the theory of quadratic forms. GAUSS now proceeds as follows: I. . pq / D . pq / for primes satisfying the usual conditions and where, in addition, / D 1. Choosing the sign of p  1 mod 4. Assume first that . pq / D 1; then . q p q in such a way that ˙q  1 mod 4 we see that the equation ˙q D b 2  cp is solvable. Putting D D ˙q, then .p; b; c/ is a form of the first kind with determinant D  1 mod 4. Since D is a prime of the form 4n C 1, the number of possible characters is 2. According to our lemma there is only one genus, the principal genus. Thus .p; b; c/ can be transformed into the form .1; 0; D/, and since . D1 / D 1 this implies . pq / D C1. If . pq / D 1, then we must have . pq / D 1. In fact, if we had . pq / D C1, then there would exist a form .q; b; c/ of the first kind with determinant p  1 mod 4, which would imply . pq / D C1, and this contradicts the assumption that . pq / D 1. II. . pq / D . pq / where both primes have the form 4n C 3. In this case GAUSS considers forms of determinant D D pq  1 mod 4. The number of possible characters is 4, and by our Lemma there exist at most 2 genera for D. The two forms of the first kind .1; 0; pq/ and .1; 0; pq/ belong to distinct genera, the first one to the principal genus. Thus the form .p; 0; q/ must belong to a genus represented by one of the two forms above. If .p; 0; q/ belongs to the principal genus, then . pq / D C1, . q / D 1, hence . pq / D 1. If .p; 0; q/ belongs to the p / D 1, hence . pq / D 1. This genus represented by .1; 0; pq/, then . pq / D 1, . q p proves the reciprocity law.

1 Preliminary Remarks

65

1.2 Kummer’s First Proof [48] Assume that in the Pell equation t 2  Du2 D 1

(6.3)

D has the form 4n C 1; then t is odd and u even. From .t C 1/.t  1/ D Du2 we get (

t C1

D 2m 2

t 1

D 2m0 2

mm0 D D  ; 2 D u

(6.4)

and subtracting one from the other we find 1 D m 2  m0 2 :

(6.5)

Now let t and u be the minimal natural numbers satisfying (6.3), then there is only one decomposition in (6.4), and the pairs .m; D/ D .1; m0 / or .m0 ; 1/ are excluded since  and  are smaller than t. From 1 D m 2  m0 2 we deduce the important relations m m0

D 1;

 m0  m

D 1;

 m0  p

D 1;

(6.6)

where p is an arbitrary factor of m. Now KUMMER factors D into prime factors in several different ways. I. D D pp0 and p  p 0  3 mod 4. Then Eq. (6.5) is (after the exclusion of the forms 1 D  2  pp0 2 and 1 D pp0  2  2 , which are impossible according to the above remarks) only solvable for the following two forms: 1 D p 2  p 0 2 if 1 D p 0  2  p2 if

p p0  p0  p

D 1; D 1;

 p0  p p p0

D 1; D 1I

this shows that if if

p p0 p p0

D C1; then D 1; then

 p0  p  p0  p

D 1; and D C1:

66

6 Proofs Based on the Theory of Quadratic Forms

II. D D pp0 q, p  p 0  3 mod 4, and q  1 mod 4. Then D can be factored in 23 D 8 different ways into two factors; after the exclusion of the two cases m D 1 and m0 D 1 according to the above assumption there remain six cases to be distinguished. Fixing p 0 in such a way that  p0  p

D 1;

i.e.,

p p0

D C1;

and

 p0  q

D 1;

and excluding those of the six cases contradicting these assumptions there remain the following three possibilities: 1 D p 2  p 0 q2 if

p 

D 1;

q

D 1I q p p  q q 1 D q 2  pp0 2 if D 1; D 1; D 1I p p0 q p q q D 1; D 1; D 1: 1 D pp0  2  q2 if p p0 q

(6.7) (6.8) (6.9)

If . pq / D 1, then the first of the three cases must hold, and it follows that . pq / D 1. Similarly, if . pq / D 1, then we are in the third case, and must have . pq / D 1. This means3 : p  q if D C1; then D C1; and q p p  q D 1; then D 1: if p q For finding out what happens if . pq / D 1, we choose p 0 in such a way that p p0

D C1;

and

q p0

D 1: 0

By the cases of the reciprocity law already proved we know that . pp / D 1 and 0

. pq / D 1, and now only two cases remain: 1 D p 2  p 0 q2 if

p 

D 1;

q

D 1I q p p  q D 1; D 1: 1 D pp0  2  q2 if q p

(6.10) (6.11)

3 [FL] At this point, BAUMGART erroneously thinks that the second case below (omitted in his thesis but given by KUMMER ) is superfluous.

1 Preliminary Remarks

67

This implies p 

q

D 1; and q p p  q  D C1; then D C1: if p q if

D 1; then

III. D D pp0 qq0 , p  p 0  3 mod 4, and q  q 0  1 mod 4. K UMMER assumes that there exist primes p and p 0 such that p q

D

 p0  q0

D 1

and

p q0

D

 p0  q

D C1:

From the 16 possible factorizations of D we can exclude those contradicting these conditions as well as the two cases in which m or m0 equals 1; then it is easily seen that (6.5) must have one of the following 3 forms: 1 D pq 2  p 0 q 0 2 if 1 D p 0 q 2  pq0 2 if 1 D p 0 q 0  2  pq2 if

p p0  p0  p  p0  p

D 1;

q

D 1; D 1;

D 1;

q0 q

q0  q0  q

 q0  q

D 1;

D 1;

 q0 

q q0

D 1I

q

(6.12)

D 1I

D 1:

(6.13) (6.14)

0

Now if . qq / D 1, then only the second case is possible, and then we must have . qq0 / D 1; if . pq / D C1, then we must be in the first or the second case, and in both cases we have . pq / D C1. The three theorems we have proved and which can be formulated as follows: if

p

D C1; then

 p0 

D 1I

p0 p q p  D C1; then D C1I if q p  q0  q D 1I D 1; then if q0 q

p

 p0 

D C1: p0 p p  q if D 1; then D 1: q p q  q0  if D C1: D C1; then q0 q if

D 1; then

(where p  p 0  3 mod 4, q  q 0  1 mod 4), can be combined into the wellknown fundamental theorem in the theory of quadratic residues and nonresidues. In this proof we used the assumption that, given a positive odd prime r and a prime q  1 mod 4, there exist primes p of the form 4n C 3 with . pr / D 1 and

68

6 Proofs Based on the Theory of Quadratic Forms

. pq / D C1. It is easy to see that this assumption is satisfied if it can be shown4 that every arithmetic progression in which the first term and the common difference are coprime represents infinitely many primes. This was proved by DIRICHLET [9], and thus the assumption is seen to be true.

1.3 Kummer’s Second Proof [48] Let p and p 0 be distinct positive primes of the form 4n C 3, and q and q 0 positive primes of the form 4n C 1. If r is a prime that can be represented by a binary quadratic form C of determinant p, then  p  r

D 1;

(6.15)

and in general the class C to which this form belongs will not be the principal class; but in any case some power of r will be represented by K D x 2 C py2 , and the exponent of r will be an odd integer and a divisor of the class number n of the quadratic forms of determinant p. In fact, the classes K, C , C 2 , . . . , C  all belong to the principal genus, and for sufficiently large  they cannot all be distinct. Now if C r D C s for some integers r > s, then C rC1s D C . Setting r  s D m  1 we either have m D n (and hence n  0 mod m) or m < n. In the first case, K; C; C 2 ; : : : ; C m1

(6.16)

completely exhaust the principal genus; in the second case, this does not happen. Now if C 0 is a class not contained in C , . . . , C m1 , K, then C 0 ; CC0 ; C 2 C 0 ; : : : ; C m1 C 0

(6.17)

are m pairwise distinct forms none of which occurs in (6.16). Thus again we either have 2m D n or 2m < n. In the first case the claim that n  0 mod m is true; in the second case it is not.5 Now we introduce a new class C 00 , and if this is not sufficient, another one etc. Thus we obtain the result that m divides n. This exponent m is necessarily odd because for determinant p there is only one ambiguous class, and the other form classes come in pairs since C m D C  .

4

[FL]: DIRICHLET’s proof of the theorem on primes in arithmetic progression used the quadratic reciprocity law. The first proof of Dirichlet’s Theorem not using quadratic reciprocity was much later given by M ERTENS . 5 [FL] What BAUMGART means here is that in this case, the listed classes do not exhaust the principal genus.

1 Preliminary Remarks

69

Now if m D 2h C 1, then x 2 C py2 D r 2h  r, which implies . pr / D 1. Thus if / D 1, then . p r if

 p  r

D 1 then

r  p

D 1:

(6.18)

Now if r is a prime that can be represented by a form of determinant q, then . qr / D 1, and we similarly get x 2  qy2 D r 2h r; which implies . qr / D 1. Thus if

q  r

D 1 then

r  q

D 1:

(6.19)

The two formulas (6.18) and (6.19) give the reciprocity law, since r is an arbitrary [positive] prime, p a prime of the form 4n C 3, and q a prime of the form 4n C 1.

Chapter 7

The Supplementary Laws of the Quadratic Reciprocity Law and the Generalized Reciprocity Law

1 The Supplementary Laws In our investigations we have made the assumption that we have already proved the supplementary laws of quadratic reciprocity, which can be expressed by the formulas .I /

 1  p

D .1/

p1 2

and .II/

2 p

D .1/

p 2 1 8

:

In this section we will verify formulas (I) and (II) using the methods that we have already used in the chapters above for deriving the relation  p  q  q

p

D .1/

p1 q1 2  2

:

First we remark that formula (I) is an immediate consequence of FERMAT’s Theorem.

1.1 Proof of Formula (I) Using “Associate Residues” (See [21, p. 135], [25, Art. 109]), for Formula (II) Using Induction The linear congruence ay  1 mod p has, for coprime integers a and p, exactly one solution. If a is an arbitrary choice of the p1 quadratic residues modulo p, then y is 2 either equal to a or different from a; in the last case, EULER calls a and y associated

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_7

71

72

7 The Supplementary Laws of Quadratic Reciprocity

residues (residua socia). If the first case occurs b times, the second e times, then p1 D b C 2c; 2 hence the number of quadratic residues a satisfying the congruence ay  1 mod p, i.e, that satisfy a2  1 mod p, is even for p D 4n C 1 and odd for p D 4n C 3; in formulas: b  0 mod 2 if p D 4n C 1; b  1 mod 2 if p D 4n C 3: The residues 1 and p  1 satisfy the congruence x 2  1 mod p, and since this congruence has at most two solutions, they are the only ones: thus b  2. Since 1 is a quadratic residue modulo all primes, this shows that p  1  1 mod p is a quadratic residue modulo p D 4n C 1 since we must have b  0 mod 2, and a quadratic nonresidue modulo p D 4n C 3 since here we must have b  1 mod 2. Formula (II) was proved by GAUSS [25, art 112 ff.] using induction. The result is true by computation e.g. for primes less than 100. Now assume that . 2t / D C1 for some prime t D 8n ˙ 3 bigger than 100, then GAUSS puts 2  a2 mod t, where a is odd and less than t; then 2 D a2 C tu for some integer u D 8n  3, and moreover . 2u / D 1. Now assume that t D 8n ˙ 3 is the minimal integer >100 with . 2t / D 1, then this contradicts . 2u / D 1 and u < t; thus we must have . 2t / D 1 if t D 8n ˙ 3. The proof for the cases t D 8n C 1; 7 is similar; all we have to do is replace 2 by 2.

1.2 Proof of the Supplementary Laws Using Reduction PETERSEN bases his proof of the quadratic reciprocity law on the half-system 1, 3, 5, . . . , p  2, i.e., admits only odd numbers less than p, and defines  in . pq / D .1/ as the number of negative odd residues less than p in q, 3q, . . . , .p  2/q. For q D 1 we immediately find that  is the number of negative odd residues in 1, 3, . . . , .p  2/, that is, we get  D p1 . 2 Similarly we have . p2 / D .1/ , where  is the number of negative odd residues less than p in 2; 2  3; 2  5; : : : ; 2.p  2/ mod p:

(7.1)

Now since 2.p  a/ C 2a  0 mod 2, we get e.g. for p D 8n C 1 that  D D 2n, i.e. . p2 / D 1. The procedure for p D 8n  1, 8n C 3, 8n C 5 is similar, except that in the first two cases the middle term in the sequence (7.1) plays a special role. 1 p1 2 2

1 The Supplementary Laws

73

1.3 Proof of the Second Supplementary Law Using Cyclotomy We have found (see Chap. 4) that GD

X

ra 

X

rb D

q .1/.p1/=2 pI

X

ra C

X

r b D 1;

which implies X

ra D

1 .1 C G/: 2

(7.2)

Since X

r ka C

X

a

r kb D 1

and

X a

b

r ka 

X b

r kb D

p  G; q

we get X

r ka D

p   1 1C G : 2 q

(7.3)

Equations (7.2) and (7.3) now imply X

ra

2



X a

r 2a 

 2  3 C .1/.p1/=2 p 1 1C G; D 4 2 p

(7.4)

which implies the second supplementary law since the right-hand side must be an integer.

1.4 Proof of the Supplementary Laws Using Quadratic Forms [25, art. 262] For determinant D D 4n C 1 D p, the form .1; 0; p/ is a form of the first kind belonging to the principal genus. Thus 1 is a quadratic residue modulo p. Assume that 1 is also a quadratic residue modulo some prime p D 4n C 3, then 1 D b 2  cp, and there exists a form .p; b; c/ of the first kind and of determinant 1. This form has character 1, which is impossible: thus 1 is a quadratic nonresidue modulo p, which is what we wanted to show. p 2 1

The proof that . p2 / D .1/ 8 does not differ substantially from the one for the first supplementary law, which is why we will omit it. Suffice it to say that one has

74

7 The Supplementary Laws of Quadratic Reciprocity

to use the form (

.8; 1; 1p 8 / .8; 3;

p1 / 8

if p  9 mod 16; if p  1 mod 16

and determinant p, and the form .p; b; c/ if p  3; 5; 7 mod 8 and for determinant 2.

2 The Generalized Reciprocity Law As we have seen, the quadratic reciprocity law can be expressed by the formulas  1 

.I /

p 2

D .1/

p1 2

D .1/

p 2 1 8

I

I p  p  q  p1 q1 D .1/ 2  2 : q p

.II/ .III/

Here p and q are distinct positive odd primes. These formulas can be generalized to negative primes; in fact, if we put p D "jpj;

q D ıjqj for "; ı D ˙1;

then (see Busche [4])  1  p 2

D .1/

"p1 2

D .1/

p 2 1 8

;

; p  p  q  "p1 ıq1 "1 ıq1 ı1 "p1 D .1/ 2  2 C 2  2 C 2  2 : q p Using JACOBI’s generalization of Legendre’s symbol (see Chap. 1), a simple calculation (cf. p. 6) shows that these three formulas remain valid for composite integers. In fact, if P and Q are coprime odd integers, and if we put P D "jP j;

Q D ıjQj for "; ı D ˙1;

2 The Generalized Reciprocity Law

75

then  1 

"P 1

D .1/ 2 ; P 2 P 2 1 D .1/ 8 ; P  P  Q  "P 1 ıQ1 "1 ıQ1 ı1 "P 1 D .1/ 2  2 C 2  2 C 2  2 : Q P P / loses its meaning. Finally, if P and Q are not relatively prime, then the symbol . Q P In this case we say that . Q / is 0. We also have to remark that GAUSS’s -Lemma was generalized by SCHERING 1 [65, 69] who shows that, if A and P are integers with P coprime to 2A, then . PA / D .1/ , where  is the number of negative minimal residues in the sequence

A; 2A; 3A; : : : ;

P 1 A mod P: 2

In the article [69], SCHERING gave an arithmetic proof for this result. Concerning SCHERING ’s generalization of GAUSS’s Lemma we should say that KRONECKER, in his article [43] published 1876, remarks that he has taught this generalization already 1869/1870 in his lectures. – Based on this generalized lemma GENOCCHI [38] has given a simple proof of formula (III). This proof is so similar to the one by the same author discussed in Chap. 3 that we may skip it here.

Notes The following list contains references to articles in which the quadratic character of small numbers is discussed. The list is probably far from being complete.

Additional References [1] G. Libri, Mémoire sur la théorie des nombres I, J. Reine Angew. Math. 9 (1832), 54–80 [2] V.A. Lebesgue, Recherches sur les nombres, Journal de Math. Pures Appl. 3 (1838), 113–144 [3] T. Schönemann, Ueber die Congruenz x 2 C y 2  1 .mod p/, J. Reine Angew. Math. 19 (1839), 93–112 [4] A.-L. Cauchy, Mem. Acad. Sci. Paris 17 (1840), 440, note IV

1

[FL] Actually this generalization had been obtained already in 1867 by JENKINS [42].

76

7 The Supplementary Laws of Quadratic Reciprocity

[5] E. Prouhet, Mémoire sur la théorie des résidus dans les proportions géométriques, Nouv. Ann. Math. 5 (1846), 652–658 [6] G. Oltramare, Considérations générales sur les racines des nombres premiers, J. Reine Angew Math. 45 (1853), 303–344 [7] V.A. Lebesgue, Introduction à la théorie des nombres, Paris 1862, p. 77 [8] M. Stern, Ueber eine der Theilung der Zahlen ähnliche Untersuchung und deren Anwendung auf die Theorie der quadratischen Reste, J. Reine Angew. Math. 61 (1863), 66–94 [9] V. Bouniakowsky, Sur les congruences binômes exponentielles à base 3 et sur plusiers nouveaux théorèmes relatifs aux résidus et aux racines primitives, Bull. Acad. St. Pétersbourg 14 (1869), 356–381 [10] T.J. Stieltjes Over het quadratische rest-karakter van het getal 2, Nieuw Arch. Wiskunde 9 (1882), 193–195 Œuvres Complètes I, 137–144 [11] T.J. Stieltjes, Bijdrage tot de theorie der derde-en vierde-macht resten, Amst. Versl. en Meded. 17 (1882), 338–417; Œuvres Complètes I, 145–209French translation: Ann. Fac. Sci. Toulouse 11 (1883), 1–65; Arch. Neerl. 18, 358–436; Œuvres Complètes I, 210–275 [12] T.J. Stieltjes, Sur le caractère quadratique du nombre 2 comme résidue ou non-résidue quadratique, Bull. Sci. Math. Paris (2), 8 (1884), 175–176 [13] G. Frattini, Carattere quadratico di 2 come conseguenza prossima del teorema di Wilson, Rom. Ist. Tecn. Ann. 8 (1883), 89–94 [14] A. Matrot Sur les residus quadratiques, Comptes Rendus Assoc. Franç. Limoges 19 (1890), 82–88 [15] J. Hacks, Ueber die Klassenanzahl der zu einer negativen Determinante D D q gehörigen eigentlich primitiven quadratischen Formen, wo q eine Primzahl von der Form 4n C 3 ist, Acta Math. 14 (1891), 321–328 [16] A. Schiappa-Monteiro, Sur un théorème relativ à la théorie des nombres, Revista Scientifica 1 (1884); see also Progreso mat. 2 (1892), 257–262 [17] T. Pépin, Formes linéaires des diviseurs de x 2 ˙ A, C. R. Acad. Sci. Paris 123 (1896), 683– 686, 737–740 [18] G. Tarry Les imaginaires de Galois, Assoc. l’Avancement Sci. 1911 (1912), 12–24 [19] L.v. Grosschmid, Bemerkungen über den quadratischen Charakter der Zahl 2 (Hungar.), Math. és phys. lapok 27 (1918), 80–90 [20] A. Cerone, Sulle formole di moltiplicazione delle funzioni circolari e teorema di reciprocita pei residui quadratici, Period. Mat. 1916 [21] R. Paolantonio, Alcune relazioni fra i coefficienti binomiali e nuove dimostrazioni che ne discendono per due teoremi aritmetici, Bollettino U. M. I. 10 (1931), 205–209 [22] C.A. Levin, En faktoruppdelning av binomet x p1  1 .mod p/ (Swedish), Elementar Mat. Fys. Kemi, Stockholm, 20 (1937), 81–89 [23] E. Storchi, Nuova dimostrazione di un teorema sui numeri primi, Periodico Mat. 18 (1939), 247–276 [24] G. Stoller, The Quadratic character of 3 in finite prime fields, Amer. Math. Monthly 71 (1964), p. 1033 [25] K.S. Williams, The Quadratic character of 2 mod p, Math. Mag. 49 (1976), 89–90 [26] J. Froemke, J. Grossman, An algebraic approach to some number-theoretic problems arising from paper-folding regular polygons, Amer. Math. Monthly 95 (1988), 289–307 [27] W. Watkins, The quadratic residues 1 and 3, Amer. Math. Monthly 107 (2000), no. 10, 934

Chapter 8

Algorithms for Determining the Quadratic Character

In the following we will present various ways of computing the symbol . ab /. Basically two methods have been applied. One is based on a direct application of the reciprocity law, and the other on a development of the fraction ab into a continued fraction. In the latter case . ab / may be computed both from the quotients and the residues that occur in the development as a continued fraction. The first method is easily explained by an example. Assume we have to compute . 365 /. Then we find 847  365  847

D

 847 

365  117   365   14   117  D D D D 365 117 117 14  5   14   1  D D D C1; D 14 5 5

since 365  1 mod 4;

hence1 . 365 / D C1. 847

1 Gauss’s Method for Determining (a/b) [32, p. 59]. GAUSS assumes that a and b are coprime and positive and then writes a D bn1 C c; b D cn2 C d;

1 [FL] BAUMGART is working without comment with even denominators. I do not know what 14 / D . 14 / by EULER ’s version of quadratic he was thinking here. Observe, however, that . 117 5 reciprocity since 117  5 mod 4  14.

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_8

77

78

8 Algorithms for Determining the Quadratic Character

::: f D gn C 1: If . ba / D .1/, then the system of equations2 1  D .a; b/ D a0 b 0  n1 .b 0 b 0 C b 0 /  .b; c/; 2 1 .b; c/ D b 0 c 0  n2 .c 0 c 0 C c 0 /  .c; d /; 2 ::: 1 .f; g/ D f 0 g 0  n .g 0 g0 C g 0 /  0; 2 where we have put x 0 D b x2 c, implies that  D a0 b 0  b 0 c 0 C c 0 d 0  : : : ˙ f 0 g 0 1  fn1 .b 0 b 0 C b 0 /  n2 .c 0 c 0 C c 0 / ˙ : : : ˙ n .g 0 g 0 C g 0 /g: 2 This formula is not really suited for actual calculations because it is quite complicated. The algorithm of SYLVESTER is based on the same idea as GAUSS’s algorithm above. He writes a D bn1 C "1 c1 ; b D c1 n1 C "2 c2 ; ::: where the quotients, which were arbitrary in GAUSS’s method, are chosen to be the nearest even integer to ba ; this means that we n can be bigger or smaller than ab . The remainders c then all are odd (and form the chaine impaire of SYLVESTER3 ). Now we find

P b D .1/  f.c 1/.c1 1/C." 1/."1 1/g : a

2

[FL] Here  D .a; b/ actually means   .a; b/ mod 2.

3

Cf. GEGENBAUER [35].

2 The Algorithms by Eisenstein [13] and Lebesgue [51]

79

195 As an example we take . 1;901 /. Then

1;901 D 195  10  49; 195 D .49/  .4/  1; which means that c D 1, c1 D 49, and " D "1 D 1. This implies 195 . 1;901 / D 1.

2 The Algorithms by Eisenstein [13] and Lebesgue [51] In the formulas by GAUSS and SYLVESTER, the quotients of continued fractions play a prominent role. The residues in GAUSS’s method are arbitrary, even or odd, positive or negative; Sylvester only considers odd remainders, positive or negative.

2.1 Eisenstein Eisenstein admits only positive odd remainders in his algorithm. Thus he finds a D bn1 C "1 c; b D cn2 C "2 d; .a > b > : : : ; "k D ˙1/

::: f D gn C " :

Attaching to every such equation the number 1 if divisor and remainder have the form 4n C 3, and 0 if one of them or both have the form 4n C 1, then . ab / D 1 if the sum of these numbers is even, and . ab / D 1 if it is odd.

2.2 The Algorithms of Lebesgue The algorithms of Lebesgue differ from E ISENSTEIN’s in the fact that LEBESGUE only uses even remainders. If a D bn1 C 2m1 "1 c; b D cn2 C 2m2 "2 d; ::: f D gn C 2m " ;

80

8 Algorithms for Determining the Quadratic Character

then the first observation is this: if  denotes the number of equations with a divisor of the form 8n ˙ 3 and a remainder with a factor 2mi with odd exponent mi , and  the number of equations in which the number on the left and the remainder (stripped from the power of 2) are both of the form 4n C 3, then . ab / D .1/C . Second, if  denotes the number of odd exponents m corresponding to divisors of the form 8n ˙ 3,  the number of odd exponents corresponding to dividends of the same form, and finally  the number of factors 4n  1 corresponding (after omitting factors 2 and ") to remainders of the form 4n  1, then . ab / D .1/CC . In order to show these formulas at work we use an example by L EBESGUE. /. 1. E ISENSTEIN’s Method. x D . 2;933 3;785 3;785 D 2;933  2  2;081;

279 D 181  2  85I

2;933 D 2;081  2  1;229;

181 D 85  2  11I

2;081 D 1;229  2  377;

85 D 11  8  3I

1;229 D 377  4  279;

11 D 3  4  1I

377 D 279  2  181; / D 1. and this implies . 2;933 3;785 2. LEBESGUE’s First Method: 3;785 D 2;933  1 C 4  213; 2;933 D 213  13 C 4  41; 213 D 41  5 C 8;

 D 0I  D 0I

/ D 1. and again we find . 2;933 3;785 3. LEBESGUE’s Second Method: 3;785 D 2;933  1 C 852;

98 D 83  1 C 15I

2;933 D 852  3 C 377;

83 D 15  5 C 8I

852 D 377  2 C 98;

15 D 8  1 C 7I

377 D 98  3 C 83;

8 D 7  1 C 1:

This implies  D 0,  D 1 and  D 1, hence . 2;933 / D .1/2 D C1. 3;785 Remark If a remainder ˙2m r 2 occurs, all subsequent calculations are superfluous.

3 The Algorithms by GEGENBAUER [35]

81

3 The Algorithms by GEGENBAUER [35] Whereas the remainders in the algorithms of GAUSS are arbitrary, odd in E ISENSTEIN ’s and even in L EBESGUE ’s method, G EGENBAUER used even and odd remainders alternately. If a and b are coprime odd integers with a > b, then GEGENBAUER develops 2b a into a continued fraction whose partial numerators are all 1 and whose partial denominators are even. The remainders then are alternatively even and odd; if their sign is ", then b  a

1

D .1/ 8 f2

P

 " "1 C.a1/

2g

:

P 2 Thus b is a quadratic residue modulo a if " "1   ."1/ mod 8, and b is a 2 P ."1/2 quadratic nonresidue modulo a if " "1  4  2 mod 8. In other words: . ba / D C1 if the difference of the number of pairs with equal signs in the sequence " and the number of pairs with different signs is congruent to 0 or 6 modulo 8; and . ba / D 1 if this difference is congruent to 2 or 4 modulo 8. In fact, the term " "1 is positive if "1 and " have equal sign, and negative is their signs are different. /: Example Determine4 x D . 173 913 346 D 913  0  346; 913 D 346  2  221;

96 D 29  4  20I 29 D 20  2 C 11I

C346 D 221  .2/  96;

20 D 11  2  2I

C221 D 96  .2/ C 29;

11 D 2  6 C 1:

The number of pairs with equal signs is 1, the number of sign changes 5, hence . 173 / D 1. 913

3.1 The Second Method The second method for determining . ab / given by GEGENBAUER starts by devela oping 2b , where a and 2b are relatively prime, into a continued fraction whose partial numerators are equal to 1, and whose partial denominators are odd; then the remainders are again alternatively even and odd. By arguments completely analogous to those above G EGENBAUER shows that . ba / D C1 if the difference between the number of pairs of residues with equal signs and the number of pairs

4 This example is due to G EGENBAUER , whose computations contain an error, however: instead of 96 W 29, he writes 96 W 27 etc.

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8 Algorithms for Determining the Quadratic Character

with different signs is congruent to 1 or 7 modulo 8, and that . ba / D 1 if this difference is congruent to 3 or 5 modulo 8.

4 An Algorithm by Kronecker 1 For determining . n / for integers with jn0 j > jn1 j, K RONECKER writes n0

n0 D 2r1 n1  n2 ; n1 D 2r2 n2  n3 ; ::: n 2 D 2r 1 n 1  1; denote the number of where all the n are odd and jnk j > jnkC1 j. If  and pairs .nk ; nkC1 / with equal resp. different sign, and if  0 and 0 denote the cor0 1 responding quantities for the sequence of nk mod 4, then . n / D .1/. /=2 D n0 0 .1/.  /=2 . 105 1 Example . n n0 / D . 143 /. The algorithm is:

143 D 2  105  67

9 D 2  .2/7 C 5

105 D 2  67  29

7 D 2  .1/.5/  3

67 D 2  29 C 9

5 D 2  .1/e C 1

29 D 2  .2/.9/  7 Then our sequence of integers n is 1; 143; 105; 67; 29; 9; 7; 5; 3; 1: The corresponding sequence of residues modulo 4 is 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; hence  D  0 D 4 and

D

0

D 5. This implies

105 143

D 1:

Part II

Comparative Presentation of the Principles on Which the Proofs of the Quadratic Reciprocity Law Are Based

Chapter 9

Gauss’s Proof by Induction

As we have already seen in Chap. 2, GAUSS distinguishes eight cases in his first proof. This makes the first proof so long that it hardly can be found useful for the proof of such a simple law. Yet this lack of shortness is not so much a consequence of the principle of induction on which the proof is based but rather of the notation. In fact, GAUSS writes pRq instead of . pq / D C1 and pN q for . pq / D 1. This forces him to distinguish eight cases, and this could have been avoided through the use of LEGENDRE ’s symbol. Actually we have seen that by using Legendre’s notation the eight cases distinguished by GAUSS can be reduced to two. DIRICHLET [11] has first pointed out this drawback of GAUSS’s first proof and presented it using the L EGENDRE-JACOBI symbol. We will follow him here. After this remark referring to the purely formal side of the proof we now address its nature. The general impression is first a great satisfaction because the prove “nowhere leaves the area of quadratic congruences” (see [11]). All other proofs, however short and elegant they may be, lack this simplicity. GAUSS himself (see [26, p. 70] or [30, p. 4]) says about his first proof: Sed omnes hae demonstrationes, etiamsi respectu rigoris nihil desiderandum relinquere videantur, e principiis nimis heterogenis derivatae sunt, prima forsan excepta quae tamen per ratiocinia magis laboriosa procedit, operationibus proxilioribus premitur.

The fundamental principle of our proof is the principle of induction. The fact that the reciprocity law holds for the two smallest odd primes 3 and 5 led GAUSS to the ingenious idea to prove the law by going from the numbers 3 and 5 successively up to larger and larger primes. This idea had to be formulated first in order to make mathematical deductions from it possible. This was accomplished by G AUSS by the following conclusion: If the reciprocity law holds for all primes below q, if p and p 0 are two such 0

p1 p 0 1

primes < q (thus . pp0 /. pp / D .1/ 2 2 ), and if this implies the correctness of the fundamental theorem for p and q (or p 0 and q), then the reciprocity law is proved in general because it holds for 3 and 5. It turned out, however, that there © Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_9

85

86

9 Gauss’s Proof by Induction

is an obstacle to the proof, which forced GAUSS to distinguish eight cases. In fact, the proof depends so intimately on the properties of p and q that the difference of their properties requires the introduction of different methods. As we have already remarked, the introduction of suitable notation reduces the number of necessary cases from eight to two. These are: 1. If q and ˛ < q are arbitrary odd primes, q positive, ˛ positive or negative, and if ˛1 q1 . ˛q / D C1, then it must be shown that . ˛q /. ˛q / D .1/ 2 2 : 2. If q D 4n C 1 and p < q are arbitrary odd positive primes with . pq / D 1, then it must be shown that . pq / D 1. The verification of the first claim was relatively easy for GAUSS because the assumption . ˛q / D C1 gave him something to work with, namely the congruence x 2  ˛ mod q. The introduction of the auxiliary number f and the use of its easily presented properties led straight to the goal. If e denotes the even root of the congruence x 2  ˛ mod q, then the auxiliary number f is defined by e 2 D ˛ C fq:

(9.1)

The distinction of the two cases (a) e and f are coprime and (b) e and f are divisible by ˛ yields, using our general assumption, the reciprocity law in a simple way. The second claim was a lot harder to prove, and only after thinking laboriously for a whole year about this problem Gauss overcame all obstacles (on April 29, 1796). “G AUSS himself recorded the date of this discovery, and did the same with his other great creations” (SCHERING [66, p. 4]). The difficulty in question lies with the fact that the assumption . pq / D 1 cannot be formulated mathematically,1 since the impossibility of x 2  p mod q cannot be represented by a formula that is intimately connected with this congruence. This fact made a lemma necessary, whose formulation and proof required GAUSS’s brilliance. KRONECKER [43] called the proof of this lemma a “Kraftprobe GAUSSschen Geistes”.2 Here is this lemma: there always exists a positive odd prime p0 < q of which q is a quadratic nonresidue. For completeness sake we remark that this result not only holds for q D 4n C 1 but also if q has the form 4n C 3 [43]. For q D 8n C 5 the theorem is immediately evident, but not for q D 8n C 1. If, in this case, q were a quadratic residue of all primes less than 2m C 1 (< q), and if k denotes a root of the congruence k 2  q mod M for M D .2n C 1/Š, then we would have .k 2  1/    .k 2  m2 / D .q  1/.q  22 /    .q  m2 / mod M;

1

I.e., in a positive way.

2

A showdown of GAUSS’s intellect.

9 Gauss’s Proof by Induction

87

hence q.q  12 /.q  22 /    .q  m2 / .2m C 1/Š would have to be an integer. The impossibility of this integrality disproves the p assumption and shows that there always is a prime p0 < 2 q C 1 of which q is a quadratic nonresidue. Thus we have . pq0 / D 1. In order to show that . pq / D 1 it suffices (using the fact that if the reciprocity law holds for primes, it also holds for composite numbers) to show that . ppq 0 / D C1. As we can see the lemma was only necessary to turn the criterion . pq / D 1 into a positive statement. We now introduce an auxiliary number f by setting e 2 D pp0 C fq;

(9.2)

where e denotes the even root < q of the congruence x 2  pp0 mod q. In the course of the proof, everything depends on the behavior of e and f with respect to p and p 0 . In fact, different methods are required according as e and f are coprime to p and p 0 , p or p 0 , or divisible by p and p 0 . This part does not contain essentially new ideas. GAUSS’s first proof thus relies essentially on the properties of the integers f and f 0 in (9.1) and (9.2). These two equations are not essentially different. Equation (9.1) results from putting p 0 D 1 in (9.2). The central point of the proof are these equations, or, since (9.1) is a special case of (9.2), Eq. (9.2), and therefore the lemma claiming the existence of an odd prime p 0 < q of which q is quadratic nonresidue.

Notes GAUSS’s first proof of the quadratic reciprocity law was rarely included in textbooks; among the few exceptions are MATHEWS [Mat], AIGNER [Aig] and SHAPIRO [Sha]. LEHMER [Leh] published an English translation. The presentation by BROWN [Bro] is close to the original; CARLITZ [Car] streamlined GAUSS’s proof even further by showing that all cases can be handled simultaneously. PÉPIN [Pep] showed how to use Gauss’s Lemma on the existence of his auxiliary primes to fill the gaps in LEGENDRE ’s proof. TEEGE [Te1, Te2] gave an analytic proof of the existence of Legendre’s auxiliary primes, as did ROGERS [Rog]. KÖNIG [Kön] presented a version of GAUSS’s induction proof that did not require any auxiliary primes at all; L IPKE [Lip] and BACHMANN [Ba, p. 402] observed that KÖNIG’s proof contained a gap, and so does the exposition of KÖNIG’s proof in PIEPER’s book [Pie]. TATE noticed that GAUSS’s first proof could be applied to the calculation of K2 Q (see BIRCH [Bir]).

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9 Gauss’s Proof by Induction

Additional References [Aig] [Ba] [Bir] [Bro] [Car] [Kön] [Leh] [Lip]

[Mat] [Pep] [Pie] [Rog] [Sha] [Te1]

[Te2]

A. Aigner, Zahlentheorie,de Gruyter, 1975 Bachmann, Niedere Zahlentheorie. Erster Teil, Leipzig 1902; reprint Chelsea 1968 B. J. Birch, K2 of global fields, Institute on Number theory, Proc. Symp. Pure Math. XX, Amer. Math. Soc. (1971), 87–95 E. Brown, The first proof of the quadratic reciprocity law, revisited, Amer. Math. Monthly 88 (1981), 257–264 L. Carlitz, A note on Gauss’ first proof of the quadratic reciprocity theorem, Proc. Amer. Math. Soc. 11 (1960), 563–565 J. König, Das Reciprocitätsgesetz in der Theorie der quadratischen Reste, Acta Math. 22 (1899), 181–192 D.H. Lehmer, Gauss’ first proof of the law of quadratic reciprocity, in: D. E. Smith, A source book in mathematics (1959), 112–118 J. Lipke, Ein Fehler im König’schen Beweise des Reciprocitätsgesetzes in der Theorie der quadratischen Reste, lecture at the February meeting of the San Francisco section of the AMS, February 29, 1908; cf. Bull. Amer. Math. Soc., May 1908, 359–360 G.B. Mathews, Theory of numbers, Cambridge 1892; 2nd ed. New York 1961 T. Pépin, Dissertation sur deux démonstrations du théorème de réciprocité de Legendre, Rom. Acc. Pont. d. Nuovi Lincei 51 (1898), 123–144 H. Pieper, Variationen über ein zahlentheoretisches Thema von Carl Friedrich Gauss, Birkhäuser Verlag Basel-Stuttgart, 1978, 183 pp K. Rogers, Legendre’s theorem and quadratic reciprocity, J. Number Theory 6 (1974), 339–344 H. Shapiro, Introduction to the theory of numbers, Wiley & Sons 1983 H. Teege, Über den Legendreschen Beweis des sogenannten Reziprozitätsgesetzes in der Lehre von den quadratischen Resten und seine Vervollständigung durch den Nachweis, dass jede Primzahl von der Form 8n C 1 quadratischer Nichtrest unendlich vieler Primzahlen von der Form 4n C 3 ist, Mitt. Math. Ges. Hamburg 5 (1920), 6–19 H. Teege, Richtigstellung eines früheren Beweises für den Satz, daß es für jede Primzahl p von der Form 4n C 1 unendlich viele Primzahlen von der Form 4n C 3 gibt, von denen p quadratischer Nichtrest ist und Herleitung des Satzes, daß mindestens eine unter ihnen kleiner als p ist, Mitt. Math. Ges. Hamburg 6 (1924), 100–106

Chapter 10

Proofs by Reduction

In Chap. 3 we have reproduced twelve proofs, which are all based on one and the same lemma that we will now briefly derive in its general form. If ak D a1 ; a2 ; : : : ; a.q1/=2

.ak < q/

represents an arbitrary half-system modulo q, then ak p will also give a half-system modulo q. The ak p are not related to the ak and may coincide with them or differ from them. We would like to visualize this. The complete system of residues modulo q is apparently represented by the numbers contained in the following two columns: I: a1 a2 ::: a.q1/=2

II: a1 a2 ::: a.q1/=2

The residues pak will appear in both columns, but not in the same row, because two residues pak are never congruent modulo q. In fact, if ak p  ak 0 p mod q, then .ak  ak 0 /p  0 mod q or, since p and q are primes, ak  ak 0  0 mod q; but this is impossible since ak and ak 0 are incongruent modulo q. Now if  residues pak appear in the second column, then we get p

q1 2

a1    a q1  .1/ a1    a q1 mod q; 2

2

or p

q1 2

 .1/ mod q 



p and D .1/ : q

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_10

89

90

10 Proofs by Reduction

This is the lemma on which all proofs of Chap. 3 are based. In this way we have reduced the original criterion . pq /  p .q1/=2 mod q to . pq / D .1/ . Only for this reason I call the proofs based on this lemma “proofs by reduction”. – Thus the symbol . pq / is defined by . pq / D .1/ , where  is the number of residues in pa1 , pa2 , . . . , pa.q1/=2 that are not congruent modulo q to a1 , a2 , . . . , a.q1/=2 . In the proofs of the quadratic reciprocity law, several different half-systems may be (and have been) used. Some proofs are based on half-system consisting of positive or negative absolutely smallest remainders, others use half-systems of even and yet others of odd numbers below the corresponding prime. This is the first point in which the proofs by reduction differ. Just as  is the characteristic number of p with respect to q, there is an analogous number , the characteristic number of q with respect to p, such that . pq / D .1/ . Thus we see that in order to prove the reciprocity law we have to determine the sum  C  of the difference   . This can be done by determining  and  separately, or by computing their sum or difference directly. This is a second point in which proofs by reduction may (and do) differ. Moreover,  and  may be decomposed as  D c C 0 and  D c 0 C  0 , where c and c 0 are constants – in most cases multiples of 2 – and where 0 and  0 are numbers “displaying the property of reciprocity in an easily recognizable form” (see SCHERING [67, p. 21]). This decomposition of  and  was used in some proofs. These are the three main points in which proofs by reduction differ. We have made these remarks for the general orientation and now proceed to a careful investigation of the proofs. We begin with GAUSS’s third proof. GAUSS bases this proof on the half-system of positive absolutely smallest residues, that is, on the numbers 1, 2, . . . , q1 modulo 2 the positive odd prime q. Then  is the number of negative absolutely smallest residues in p; 2p; : : : ;

q1 p 2

mod q:

Now GAUSS defines b yx c as the greatest integer contained in theorems on such expressions b yx c, .q1/=2 



X kD1

x y

and finds, using

 X 2kp kp 2kp 2  mod 2: q q q k

Assuming p < q, which we can do without loss P of generality since the primes p c occur more than once. The and q are distinct, we see that some terms in b pk q determination of the number of terms occurring more than once will lead to a proof of our theorem. GAUSS transforms the expression  D f .p; q/ into  D f .q; p/ C c, where c is the sum of an even integer and p1  q1 . 2 2 VOIGT, an insurance agent from Swabia who died young, proceeds similarly: he also uses a half-system consisting of positive smallest absolute values and continues

10 Proofs by Reduction

91

as follows: if b kp c D h  1, then kp will give rise to a negative absolutely smallest q residue modulo q if .h  12 /q < kp < hq. Conversely, such numbers k – whose h 1

cardinality by the way is b hp c  b p 2 qc – satisfying the inequality above give rise q to negative absolutely smallest residue modulo q. Thus we find

2  X hp p1



hD1

since the maximal value for h is 

X q h

2

p1 . 2

q



h  12  q p

 ;

Applying theorems on bxc we get

 rh mod 2;



hq hq ; rh D  p p

from which LEGENDRE’s formula is easily derived. The difference and GAUSS’s is that GAUSS transforms P between VOIGT ’s proofq1  into   k b kp c mod 2 (k D 1; : : : ; 2 ) and then shows that the number of q hq hC1 b kp q c having the same value h equals b p qc  b p c. Summation from h D 1 to

p1 2

then gives  D f .p; q/  p1  q1  f .q; p/ mod 2, our well known formula. 2 2 VOIGT, on the other hand, immediately determines the number of kp giving negative absolutely smallest residue modulo q, and finds that their number for given h equals h 1

b hp c  b p 2 qc, where h  1 D b kp c. Summation over h then also gives the desired q q result. Although GAUSS’s third proof, which we just have discussed, is short and elegant, it does not seem to have completely satisfied its author, possibly because one prime is preferred to the other. The same idea that later gave rise to the introduction of determinants probably made GAUSS look for a new proof in which  and  are not determined separately, but in which their sum modulo 2 is determined directly. And GAUSS found his fifth proof which is free from the deficiency of the third proof just mentioned. The residue systems on which GAUSS’s fifth proof is built are 1, 2, . . . , p1 and 2 q1 1, 2, . . . , 2 . For the determination of the sum  C , the sequence 1; 2; : : : ; pq  1

(10.1)

is introduced, and it is assumed that p < q. The numbers in (10.1) have different properties with respect to the moduli p and q. In fact, consider an arbitrary element of this sequence; then it may give a positive or negative absolutely smallest residue modulo p or q, or modulo p and q, or it might be a multiple of p or q (but never of p and q). Using these properties then leads to a proof of the fundamental theorem. Let .s/rR (s D 1; 2; : : : ; pq1 2 ) denote the number of positive absolutely smallest residues in A modulo pq which are congruent to positive absolutely smallest

92

10 Proofs by Reduction

residues modulo p and to negative absolutely smallest residues modulo q; then we have the formulas .S /rR D .s/Rr ;

.s/rR C .S /rR D

where S denotes one of the numbers way similar to .s/. Moreover, the p1  q1 integers 2 2 tq C Rq

p1 q1  ; 2 2

pqC1 ; : : : ; pq  1 2

.t D 0; : : : ;

p3 ; 2

Rq D

(10.2)

and where .S / is defined in a

qC1 ;:::;q 2

 1/

are exactly the numbers s that are congruent modulo q to negative absolutely smallest residues, and they contain all numbers prq (rq D 1; : : : ; q1 ). If we 2 partition these numbers into three classes according as they are congruent modulo p to absolutely smallest positive or negative residues or are congruent to zero, then we get the formula .s/rR C .s/RR C  D

p1 q1  ; 2 2

(10.3)

where  is the number of prq that are congruent modulo q to negative absolutely smallest residues. Finally, using the p1  q1 numbers 2 2 p C Rp

. D 0; : : : ; q3 ; 2

Rp D

pC1 ;:::;p 2

 1/

we can similarly derive the formula .s/Rr C .s/RR C  D

p1 q1  ; 2 2

(10.4)

where  denotes the number of qrp that are congruent modulo p to negative absolutely smallest residues. The formulas (10.2)–(10.4) now imply 2.s/RR C  C  D

p1 q1  : 2 2

In this proof, the numbers  and  are split into sums of three terms; we will find a similar decomposition in BOUNIAKOWSKI’s proof. Thus GAUSS’s fifth proof is based essentially on the decomposition and enumeration of the numbers tq C Rp

and p C Rp

10 Proofs by Reduction

93

contained in the sequence 1, 2, . . . , pq  1, and on the validity of the formula .s/Rr C .S /rR D p1  q1 . It is simple and elementary in that no other auxiliary means are 2 2 necessary except for the auxiliary sequence 1, 2, . . . , pq  1. Moreover, as we have already remarked, in contrast to Gauss’s third proof, the primes p and q enter in a symmetric way. The third proof was found in 1808, the fifth in 1818. Thirty years later (1847), E ISENSTEIN published his geometric proof of the fundamental theorem, which essentially is GAUSS’s third and fifth proof translated into the language of geometry. According to GAUSS’s Lemma we have

p D .1/ ; q

2 X kp q1

D

kD1



q D .1/ ; p

q

2 X hq

;

p1

D

hD1

p

;

which implies that C 

X  kp q



hq C p

 mod 2:

and yp are EISENSTEIN considers a line yp D xq in the cartesian plane. Then xq p q the x- and the y-coordinate of the point .x; y/ on this line. For x D h and y D k, kp we see that b hq p c and b q c are respectively the number of lattice points (points with integral coordinates) with coordinates X  kp q

C

hq p

hq p

and

kp . q

 D

It is now clear from the picture that

p1 q1  : 2 2

y

yp q

xq p

x

94

10 Proofs by Reduction

In EISENSTEIN’s proof, GAUSS’s expressions b hq c and b kp c correspond to p P kp q P hq certain points on lines, and the expressions b p c and b q c to points on several such lines. Counting the number of integers with certain properties in GAUSS’s fifth proof corresponds to the enumeration of points using geometry. EISENSTEIN visualized the abstract notion of number in GAUSS’s proof by introducing geometric lengths. The arithmetic transformation of  D f .p; q/ into  D f .q; p/ C c in GAUSS’s third proof finally becomes obvious. In 1852, GENOCCHI came up with a new idea that was exploited and made fertile by SCHERING and KRONECKER. In order not to interrupt the flow of the exposition, let us return to this idea later. We now turn first to the point of view that can be seen for the first time in the proof by STERN, and which was also used by ZELLER and PETERSEN. With the help of this criterion by STERN, GAUSS’s third proof may be if not simplified, then, as the articles by ZELLER and PETERSEN show, at least be shortened. Instead of counting numbers with certain properties modulo p or q other principles are used. S TERN’s criterion is the following: If we assume that p < q and use the same half-systems, then in the sequences 1q; 2q; : : : ; hq; : : : ;

p1 q mod p 2

1p; 2p; : : : ; kp; : : : ;

q1 p mod q 2

and

no residue hq mod p occurs as kp mod q; and conversely, if p0 is the largest remainder contained in hq mod p, then no remainder kp mod q  p 0 appears in hq mod p. On the other hand, hq mod p appears in kp mod q, and kp  p 0 in hq. As we have already remarked on p. 22, the subsequent comments by STERN contain nothing essentially new and may be omitted, in particular since they are not completely correct. ZELLER bases his proof on the residue systems .1/ q; : : : ;

p1 qI 2

.2/ p; : : : ;

q1 p: 2

Assuming that p < q and considering only smallest absolute remainders, we have, according to STERN,  C  D p1 C , where denotes the number of remainders 2 in (2) that lie between  p2 and  q2 . The determination of this number is the central point of Z ELLER’s proof. The substitution k0 D

q1  k; 2

r0 D

pCq  r; 2

10 Proofs by Reduction

95

where kp  r mod q, shows that the terms kp lying between  p2 and  q2 occur in pairs, because any r 0 comes with some r, except for the terms corresponding to the limits of the substitution. In order to take care of these exceptions, consider first the case where k D 0; then k 0 p  qp mod q, hence  0 mod 2. 2 q1 0 If k D k D 4 , then we see immediately that  0 mod 2 for q  3 mod 4, mod q, giving rise to various results whereas for q  1 mod 4 we get kp  p˙q 4 depending on q D 4n ˙ 1. The proof by PETERSEN is completely analogous to this one. He uses the odd integers 1; 3; 5; : : : ; q  2 as a half-system modulo q, so  is the number of negative odd residues modulo q in p, 3p, 5p, . . . , .q  2/p. Assuming p < q again, we find  C nu D

p1 C ; 2

where denotes the number of odd residues from p, 3p, . . . , .q  2/p modulo q lying between p and q. Now let r denote the odd residues modulo q; then .2n  1/p  2mq D r;

2n C 1 D 1; 3; : : : ; q  2I

here m is chosen in such a way that q > r > q. The substitution m D n  k, p D q  2a now shows that equals the number , . . . , ˛1 q whose floor is odd. Now again arguments due to Zeller of fractions ˛q , 2q ˛ ˛ are used according to which the middle terms are important. Now we come to the proof by BOUNIAKOWSKI. This author also determines the sum  C , but decomposes  and  in a remarkable way. First he remarks that two primes must be contained in the same linear form, that is, if p D 2an C r for odd integers a; r with 1 < r < 2a  1, then q D 2an0 C r. Moreover, BOUNIAKOWSKI finds the important formula

a a1 D .1/ 2 nCm ; p where m is an integer depending on a and r, but not on n. This implies then that

a1 0 a D .1/ 2 n Cm ; q hence

a a a1 0 D .1/ 2 .nCn / : p q

96

10 Proofs by Reduction

a The beautiful formula . 2anCr / D .1/ p1 r1 D C an residues 2 2

a1 2 nCm

then follows from separating the

1; 2; 3; : : : ; p1 2 D 1; 2;

1 C a; : : : ; 1 C .n  1/a; 1 C na; 2 C a; : : : ; 2 C .n  1/a; 2 C na; ::: r1 r1 ; C a; : : : ; r1 C .n  1/a; r1 C na; 2 2 2 2 rC1 rC1 rC1 ; 2 C a; : : : ; 2 C .n  1/a; 2 ::: a; 2a; : : : ; na

into three classes as GAUSS did with the integers 1, 2, . . . , pq1 in his fifth proof. 2 The first class contains the numbers of the last row, the second the numbers of those rows that begin with r  r mod p

. D 1; 2; : : : ; a1 2 /I

finally, the third class contains the numbers of those rows that begin with a  r . The numbers of the first and the third class now are modulo p congruent to positive multiples .< p1 / of a, those of the second class to negative multiples .< p1 / of 2 2 a. If we denote the number of elements in the second class by M , then

a D .1/M : p n C m for some m depending only on a and r. Finally we find M D a1 2 Now if we set p D q C 2 a for primes p and q and an odd integer a, then we find: 



p q 2 2 a1 a1 0 nCm 2 and D .1/ D .1/ 2 n Cm ; q q p p and this implies, in light of . p2 / D .1/b

qC1 p a1 D .1/b 4 cC 2 nCm q

and

pC1 4 c

(see BOUNIAKOWSKI [3]):



p1 pC1 q a1 0 D .1/ 2 Cb 4 cC 2 n Cm : p

But this yields

p1 pC1 qC1 p q a1 0 D .1/ 2 Cfb 4 cCb 4 cgC 2 .nCn / : q p

10 Proofs by Reduction

97

These formulas then first give the remarkable decomposition of  and . Legendre’s formula now follows easily from the last equation by distinguishing the cases p  q mod 4 and p  2  q mod 4. There is a certain similarity between the proofs by BUSCHE and BOUNIAKOWSKI inasmuch as the basis of BUSCHE ’s proof is the verification that

q 2q C p

D .1/

q1 2



q : p

This formula follows immediately from BOUNIAKOWSKI’s more general

a a a1 0 D .1/ 2 .nCn / p p by putting q D r, that is, n0 D 0. As we have seen, BOUNIAKOWSKI employs GAUSS’s counting method for deriving his formula, whereas BUSCHE follows the presentation of G AUSS’s third proof. BUSCHE puts, as VOIGT before him, q3

D

2 X

h ;

hD1

where . pq / D .1/ and where h is the number of solutions k of pC1 2

kq D hp C r;

>r >p

for given h; he also puts q3

M D

2 X

Mh ;

hD1 q / D .1/M , and where Mh is defined in analogy to  as the number where . pC2q of integral solutions K of

Kq D h.p C 2q/ C r 0 ;

p1 2

C q < r 0 < p C 2q:

If we assume that q > p and that  > 0, then we find Mh D  C h ; C : hence M D  q1 2

98

10 Proofs by Reduction

Now assume that the reciprocity law holds for p and q, that is, suppose

p1 q1 q p D .1/ 2  2 I q p

(10.5)

then we find

q 2q C p



2q C p q

D .1/

q1 pC2q1 2  2

:

(10.6)

Moreover we have

" q "1 q1 D .1/ 2  2 ; q "

(10.7)

where " D ˙1. Now we can apply a lemma found by BUSCHE . It follows from the Euclidean algorithm for determining the greatest common divisor of two integers that, if the validity of some relation .x; y/ between odd coprime integers implies that .˙1; y/;

(10.8)

.x; ˙1/;

(10.9)

.x C 2y; y/;

(10.10)

0

.x; y C 2 x/;

(10.11)

also hold for integers ; 0 , then .x; y/ holds for arbitrary odd coprime integers x; y. Assuming that (10.5) holds we have derived (10.6), and (10.7) holds trivially; thus the four condition (10.8)–(10.11) are satisfied, and the quadratic reciprocity law holds in general. It remains to consider the proofs by GENOCCHI , SCHERING and KRONECKER. Their proofs – although they are analogous to those discussed above, since they also determine the sum  C  – have a special place because they – some more than others – have a certain function theoretic meaning. As we have remarked already, GENOCCHI published his proof in 1852. He considers expressions of the form u D hq  kp;

v D hq C kp 

pq1 2

.q > p; h < p2 ; k < q2 /

10 Proofs by Reduction

99

and discusses under which sonditions u and v are positive and negative, respectively. Comparing these conditions he finds that card: pos: v  card: pos: u D 0 or 1; k

k

.k D 1; : : : ; p1 / 2

according as hq is congruent modulo p to a positive or negative smallest absolute remainder. This shows that   card: pos: v  card: pos: u mod 2; h;k

h;k

and similarly   card: pos: v  card: pos: u0 mod 2; h;k

h;k

where u0 D pk  qh and h D 1, . . . ,

p1 2

and k D 1, . . . ,

q1 . 2

Thus we find

 C   card: pos: u C card: pos: u0 mod 2; h;k

h;k

which in turn immediately implies our well-known formula. SCHERING replaces the expressions u, v by the following: U D

u h k D  ; pq p q

V D

v 1 h k 1  D C  : pq 2pq p q 2

This allows him to give an exposition that is simpler than G ENOCCHI’s, and they admit a function theoretic treatment. We find



k 1 h h k  card: pos: D 0; 1 C   card: pos: k k p q 2 p q according as hq is congruent modulo p to a positive or negative absolutely smallest remainder. The other arguments are the same as GENOCCHI’s. Whereas G ENOCCHI and SCHERING assume GAUSS’s Lemma as known, KRONECKER pursues a different strategy: he replaces G AUSS ’s Lemma with a lemma from GAUSS’s first proof, hence KRONECKER’s proof stands between GAUSS’s first proof and those discussed in this section. This proof is of great importance because it connects such diverse methods. KRONECKER defines . pq / as the sign of 2 Y 2 Y h p1 q1

hD1 kD1

k :  p q

100

10 Proofs by Reduction

From this definition of . pq / the reciprocity law follows without effort. From Q k / D p1 .h  kp / we find q q

Q h .p 



P kp p D .1/ b q c : q This formula then implies the following:

p0 q





p D .1/ q 0 0

pp p p D ; q q q p0 q





p D q

q1 2



if p  p 0 mod qI

(10.12)

if p 0  p mod qI

(10.13)



q q : p p0

(10.14)

q pp 0

D

The formulas (10.12)–(10.14) show that the symbols . pq / and . pq / satisfy the same rules as the LEGENDRE-JACOBI symbols. In order to show that they coincide we therefore only have to show that . pq / D C1 if p is a quadratic residue modulo q, and that . pq / D 1 if p is a quadratic nonresidue modulo q. The first case is easily dealt with. As for the second case, by (10.12) and (10.14) we have to show that there is at least on number p for which . pq / D 1. And such a p exists indeed, as already GAUSS has partially shown in his first proof; KRONECKER completed GAUSS’s exposition. At the end of this section it remains to mention two articles by KRONECKER [45, 46] in which mainly the following results concerning the topic of this section are derived: the transformation of SCHERING’s power into a product and the direct verification that p1 q1



2 Y 2 Y h k p D sign :  q p q

hD1 kD1

The obervation that .a  x/.a  x C 12 / is negative if x lies between a and a C 12 , or q between x  12 and x, implies that, if we denote by R.a/ the fractional part of a: sign R.a/ D sign.a  k/.a  k C 12 /; where k D ba C 12 c. Now since .a k/.a k C 12 / remains positive for k ¤ ba C 12 c, we get sign R.a/ D sign

r Y

.a  k/.a  k C 12 /;

kD1

r  ba C 12 c;

10 Proofs by Reduction

101

hence sign R.qa/ D sign

r Y

.qa  k/.qa  k C 12 /;

kD1

or, assuming that q is positive,

sign R.qa/ D sign



r Y k k 1 a C ; a q q 2q

k D 1; : : : ; r 

kD1

q1 : 2

 k 0 is allowed since it only permutes the factors on the The substitution k D q1 2 right-hand side; applying it to the equation shows that



2 Y k k 1 sign R.qa/ D sign a C : a q q 2q q1

kD1

We have not made any assumptions about a so far; now we put a < moreover p and h < p2 are positive, then

qh sign p



2 Y h q1

D sign

kD1

k  p q



1 . 2

If

h k 1 :  C p q 2q

If h runs through a positive absolutely smallest half-system modulo p, then we find p1 q1





2 Y 2 Y h k 1 h k q D sign :   C sign p p q p q 2q

(10.15)

hD1 kD1

SCHERING had found the expression

k 1 h C  p q 2q hD1 kD1

 h k  card: pos: mod 2  p q

2 X 2  X



p1 q1



card: pos:

(10.16)

for  in . pq / D .1/ . Comparing (10.16) and (10.15), their similarity is immediately clear.

102

10 Proofs by Reduction

KRONECKER’s proof is, as we have seen, based on the fact that he showed, using GAUSSian observations, that the expression p1 q1



2 Y 2 Y h k p D sign  q p q

hD1 kD1

coincides with the LEGENDRE symbol. In [45], KRONECKER gives a direct derivation of this formula. According to GAUSS [26, p. 6] we have sign R.p˛0 / D .1/p˛ ; where 0 < ˛0 q2 , where k is fixed. BUSCHE finally simplifies BOUNIAKOWSKY’s proof using an elegant lemma, according to which the reciprocity law holds if it can be proved for certain special cases. Finally p we would like to mention that CAUCHY succeeded in deriving the famous sign of .1/.p1/=2 p known from the “summatio” using the transformation of theta functions.

Chapter 15

Proofs of the Quadratic Reciprocity Law

This appendix contains (a) A list of the known proofs of the quadratic reciprocity law; not included are proofs given in textbooks unless they contain a novel idea. Many of the proofs based on Gauss’s lemma as well as several others differ only marginally; nearly identical proofs were counted separately except when they were published by the same author. (b) A bibliography with references to the known proofs of the quadratic reciprocity law, together with a very brief description of the contents of the articles that were available to me.

# 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Proof Legendre Gauss 1 Gauss 2 Gauss 3 Gauss 4 Gauss 5 Gauss 6 Cauchy Jacobi Dirichlet 1 Lebesgue 1 Lebesgue 2 Schönemann Cauchy Eisenstein 1

Year 1788 1801 1801 1808 1811 1818 1818 1829 1830 1835 1838 1838 1839 1840 1844

Comments Quadratic forms; incomplete Induction; April 8, 1796 Quadratic forms; June 27, 1796 Gauss’s Lemma; May 6, 1807 Cyclotomy; May 1801 Gauss’s Lemma; 1807/08 Gauss sums; 1807/08 Gauss 6 Gauss 6 Gauss 4 N.x12 C : : : C xq2  1 mod p/ Gauss’s Lemma Quadratic period equation Gauss 4 Generalized Jacobi sums (continued)

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6_15

131

132 # 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.

15 Proofs of the Quadratic Reciprocity Law Proof Eisenstein 2 Eisenstein 3 Eisenstein 4 Kummer 1 Liouville Eisenstein 5 Lebesgue 3 Lebesgue 4 Lebesgue 5 Lebesgue 6 Schaar Plana Schaar 2 Genocchi 1 Genocchi 2 Genocchi 3 Dirichlet 2 Genocchi 4 Schaar 3 Lebesgue 7 Sylvester Kummer 2 Kummer 3 Dedekind 1 Gauss 7 Gauss 8 Jenkins Mathieu von Staudt Heime Bouniakowski Stern Zeller Zolotarev Kronecker 1 Schering 1 Kronecker 2 Mansion Dedekind 2 Dedekind 3

Year 1844 1844 1845 1846 1847 1847 1847 1847 1847 1847 1847 1851 1852 1853 1853 1853 1854 1854 1854 1860 1869 1862 1862 1863 1863 1863 1867 1867 1867 1868 1869 1870 1872 1872 1876 1876 1876 1876 1877 1877

Comments Gauss 6 Gauss’s Lemma Sine Period equation Cyclotomy Infinite products Eisenstein 2 Liouville Eisenstein 1 Lebesgue 1 Gauss’s Lemma Gauss sums Gauss 4 Gauss’s Lemma Liouville Eisenstein sine Gauss 1 Liouville Gauss 4 Gauss 7, 8 Eisenstein (geometric) Quadratic forms Quadratic forms Quadratic forms Quadratic periods; Sept. 1796 Quadratic periods; Sept. 1796 Gauss 4 Cyclotomy Cyclotomy Gauss’s Lemma Gauss’s Lemma Gauss’s Lemma Gauss’s Lemma Permutations Eisenstein sine Gauss 3 Gauss’s Lemma (signs) Zeller Gauss 6 Dedekind Sums (continued)

15 Proofs of the Quadratic Reciprocity Law # 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95.

Proof Pellet 1 Pépin 1 Sochocki Schering 2 Petersen Genocchi 5 Kronecker 3 Kronecker 4 Voigt Pellet 2 Busche Gegenbauer 1 Gegenbauer 2 Gegenbauer 3 Kronecker 5 Bork Schering 3 Schering 4 Kronecker 6 Kronecker 7 Kronecker 8 Kronecker 9 Kronecker 10 Bock Eichenberg 1 Eichenberg 2 Eichenberg 3 Hermes Lerch 1 Busche 2 Hacks Kronecker 11 Tafelmacher 1 Tafelmacher 2 Tafelmacher 3 Busche 3 Franklin Kronecker 12 Lucas Pépin 2

133 Year 1878 1878 1878 1879 1879 1880 1880 1880 1881 1882 1883 1884 1884 1884 1884 1885 1885 1885 1885 1885 1885 1885 1885 1886 1886 1886 1886 1887 1887 1888 1889 1889 1889 1889 1889 1890 1890 1890 1890 1890

Comments Stickelberger-Voronoi Cyclotomy Theta Functions Gauss’s Lemma Gauss’s Lemma Gauss’s Lemma Gauss 4 Quadratic period Gauss’s Lemma Mathieu 1867 Gauss’s Lemma Gauss’s Lemma Kronecker Schering Gauss’s Lemma Eisenstein geometric Gauss’s Lemma Gauss’s Lemma Gauss 3 Gauss 3 Gauss’s Lemma Gauss’s Lemma Gauss’s Lemma Gauss’s Lemma Schering 1 Schering 1 Schering 1 Induction Gauss 3 Gauss’s Lemma Schering Gauss’s Lemma Stern Stern/Schering Schering Gauss’s Lemma Gauss’s Lemma Gauss 4 Gauss’s Lemma Gauss 2 (continued)

134 # 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135.

15 Proofs of the Quadratic Reciprocity Law Proof Fields Gegenbauer 4 Gegenbauer 5 Gegenbauer 6 Gegenbauer 7 Gegenbauer 8 Heinitz Schmidt 1 Schmidt 2 Schmidt 3 Gegenbauer 9 Hasenöhrl Bang Mertens 1 Mertens 2 Busche 4 Lange 1 de la Vallée Poussin Lange 2 Lange 3 Hilbert Hilbert Alexejewsky Pépin 3 Pépin 4 König Lerch 2 Fischer Scheibner 1 Scheibner 2 Scheibner 3 Scheibner 4 Scheibner 5 Scheibner 6 Scheibner 7 McClintock Takagi Lerch 3 Mertens 3 Mirimanoff & Hensel

Year 1891 1891 1893 1893 1893 1893 1893 1893 1893 1893 1894 1894 1894 1894 1894 1896 1896 1896 1897 1897 1897 1897 1898 1898 1898 1899 1899 1900 1900 1900 1900 1900 1900 1900 1900 1902 1903 1903 1904 1905

Comments Gauss’s Lemma Gauss’s Lemma Gauss’s Lemma Zeller Petersen Gauss’s Lemma Gauss’s Lemma Gauss’s Lemma Gauss’s Lemma Induction Gauss’s Lemma Gauss’s Lemma Induction Gauss’s Lemma Gauss sums Gauss’s Lemma Gauss’s Lemma Gauss 2 Gauss’s Lemma Gauss’s Lemma Genus theory Cyclotomy Schering Legendre Gauss 5 Gauss 1; incorrect Kronecker 4 Resultants Zeller Kronecker ? Gauss 3 Eisenstein geometric Eisenstein sine Gauss 4 Gauss 4 Gauss’s Lemma Zeller Gauss 5 Eisenstein Stickelberger-Voronoi (continued)

15 Proofs of the Quadratic Reciprocity Law # 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175.

Proof Cornacchia Busche 5 Busche 6 Busche 7 Aubry Aubry Aubry Pépin 5 Petr Pocklington Dedekind 4 Dedekind 5 Dedekind 6 Dedekind 7 Heawood McDonnell Frobenius 1 Frobenius 2 Frobenius 3 Frobenius 4 Frobenius 5 Lasker Cerone Bartelds & Schuh Stieltjes Teege 1 Arwin Teege 2 Rédei 1 Rédei 2 Whitehead Petr 2 Skolem 1 Petr 3 van Veen Fueter Whiteman Dockeray Kapferer Scholz

135 Year 1909 1909 1909 1909 1910 1910 1910 1911 1911 1911 1912 1912 1912 1912 1913 1913 1914 1914 1914 1914 1914 1916 1917 1918 1918 1920 1924 1925 1925 1926 1927 1927 1928 1934 1934 1935 1935 1938 1939 1939

Comments Quadratic periods Zeller Eisenstein Eisenstein D Eisenstein 3 D Voigt D Kronecker Gauss 2 Mertens 3 Gauss 3 Zeller Zeller Zeller Zeller D Eisenstein 3 Cyclotomic Zolotarev Zeller Gauss 5 Gauss 3 Eisenstein 3 Stickelberger-Voronoi Eisenstein 4 Gauss’s Lemma Lattice points Legendre Quadratic forms Cyclotomy Gauss’s Lemma Gauss’s Lemma Genus theory (Kummer) Theta functions Genus theory Kronecker (signs) Eisenstein 3 Quaternion algebras Gauss’s Lemma Eisenstein 3 Liouville Gauss 3 (continued)

136 # 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215.

15 Proofs of the Quadratic Reciprocity Law Proof Dörge Rédei 3 Lewy Petr 4 Furquim de Almeida Skolem 2 Aigner Barbilian Delsarte Rédei 4 Brandt 1 Brandt 2 Brewer Zassenhaus Riesz Fröhlich Ankeny D. H. Lehmer C. Meyer 1 C. Meyer 2 Holzer Rédei 5 Reichardt Vandiver, Weaver Carlitz Kubota 1 Kubota 2 Kubota 3 Skolem 3 Skolem 4 Skolem 5 Hausner Swan 1 Koschmieder Gerstenhaber Rademacher Weil Kloosterman Chowla Burde

Year 1942 1944 1946 1946 1948 1948 1950 1950 1950 1951 1951 1951 1951 1952 1953 1954 1955 1957 1957 1957 1958 1958 1958 1958 1960 1961 1961 1961 1961 1961 1961 1961 1962 1963 1963 1964 1964 1965 1966 1967

Comments Gauss’s Lemma Gauss 5 Cyclotomy Cyclotomy Vandermonde determinants Gauss 2 Gauss 3 Eisenstein 1 Vandermonde determinants Gauss 3 Gauss 2 Gauss sums Mathieu, Pellet Finite fields Permutations Class Field Theory Cyclotomy Gauss’s Lemma Dedekind sums Zolotarev Gauss sums Cyclotomic polynomial Gauss 3 Zeller-Frobenius Gauss 1 Cyclotomy Gauss sums (Hecke) Eisenstein sine Quadratic periods Cyclotomy Finite fields Gauss sums Stickelberger-Voronoi Eisenstein, sine Eisenstein, sine Finite Fourier analysis Theta functions Holzer Finite fields Gauss’s Lemma (continued)

15 Proofs of the Quadratic Reciprocity Law # 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241. 242. 243. 244. 245. 246. 247. 248. 249. 250. 251. 252. 253. 254. 255.

Proof Kaplan 1 Kaplan 2 Kubota 4 Birch Reshetukha Agou Brenner Honda Milnor & Husemöller Zagier Allander Berndt & Evans Hirzebruch & Zagier Rogers Berndt Castaldo Springer Burde Friedlander & Rosen Frame Hurrelbrink Auslander & Tolimieri Rosen Ryan Corro Brown Cuculière Goldschmidt Kac Barcanescu Barrucand & Laubie Zantema Ely Eichler Gérardin Barrucand & Laubie Peklar Barnes Swan 2 Rousseau 1

137 Year 1969 1969 1970 1971 1971 1972 1973 1973 1973 1973 1974 1974 1974 1974 1975 1976 1976 1977 1977 1978 1978 1979 1979 1979 1980 1981 1981 1981 1981 1983 1983 1983 1984 1985 1986 1987 1989 1990 1990 1990

Comments Eisenstein Quadratic congruences Theta function K-theory (Tate; Gauss 1) Gauss sums Finite fields Zolotarev Gauss sums Weil 1964 Dedekind sums Gauss’s Lemma Gauss’s Lemma Dedekind Sums Legendre Gauss 3 Gauss’s Lemma Gauss sums Cyclotomic Gauss 3 Kronecker 3 (signs) K-theory Fourier transform Dedekind sums Gauss’s Lemma Gauss sums Gauss 1 Tate Cyclotomy Eisenstein, sine Zolotarev Stickelberger-Voronoi Brauer groups Lebesgue 1 Theta function Gauss 4 Stickelberger-Voronoi Gauss’s Lemma Zolotarev Cyclotomy Exterior algebras (continued)

138 # 256. 257. 258. 259. 260. 261. 262. 263. 264. 265. 266. 267. 268. 269. 270. 271. 272. 273. 274. 275. 276. 277. 278. 279. 280. 281. 282. 283. 284. 285. 286. 287. 288. 289. 290. 291. 292. 293. 294. 295.

15 Proofs of the Quadratic Reciprocity Law Proof Rousseau 2 Keune Kubota 5 Russinoff Garrett Motose 1 Laubenbacher, Pengelley Rousseau 3 Cornaros Young Brylinski Merindol Watanabe Ishii Beck Motose 2 Zahidi Lemmermeyer Meyer Tangedal Chapman Girstmair Hammick Murty Décaillot Luo Motose 3 Motose 4 Kim Z.W. Sun Duke & Hopkins Murty & Pacelli Szyjewski Arkhipova Robbins Kumar Kumar Kumar Castryck Gurevich, Hadani & Howe

Year 1991 1991 1992 1992 1992 1993 1994 1994 1995 1995 1997 1997 1997 1998 1999 1999 1999 2000 2000 2000 2000 2001 2001 2001 2002 2003 2003 2003 2004 2004 2005 2005 2005 2006 2006 2007 2007 2007 2008 2008

Comments Permutations Vandermonde determinants Geometry Gauss’s Lemma Weil 1964 Group algebras Eisenstein geometric Zolotarev Permutations Gauss sums Group actions Eisenstein, sine Zolotarev Gauss 4 Dedekind sums Group algebras Stickelberger-Voronoi Lebesgue 1, Ely Dedekind sums Eisenstein geometric Recurring sequences Eichler Rousseau Schur Lucas Rousseau Vandermonde determinants Vandermonde determinants Rousseau 2 Scholz Group theory Theta functions Zolotarev Gauss 4 Zolotarev Rousseau Keune Swan Lebesgue 1 Schur, Weil (continued)

15 Proofs of the Quadratic Reciprocity Law # 296. 297. 298. 299. 300. 301. 302. 303. 304. 305. 306. 307. 308. 309. 310. 311. 312. 313. 314.

Proof Kunisky Jakimczuk Schechtman Chebolu, Minac & Reis Kuroki & Katayama Hambleton & Scharaschkin Jerábek Verdure Steiner Szyjewski 2 Dicker Hambleton & Scharaschkin Karlsson Zver Baker, Shurman Demchenko & Gurevich Caldero & Germoni Burda & Kadets Brunyate & Clark

139 Year 2008 2009 2009 2009 2009 2010 2010 2010 2010 2011 2012 2012 2012 2012 2013 2013 2013 2013 2014

Comments Rousseau 2 Lebesgue 1 Gauss 4 Representations Takagi Resultants (Swan 2) Gauss 3 Elliptic curves Rousseau 2 Zolotarev Determinants Pell conics Gauss sums Dedekind sums Zolotarev Formal groups Lebesgue 1 Quadratic periods Zolotarev

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[2]

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[4]

A.M. Legendre, Recherches d’analyse indéterminée, Histoire de l’Academie Royale des Sciences de Paris (1785), 465–559, Paris 1788 LEGENDRE gives a partial proof of the “loi de réciprocité”, which was later turned into a full proof by KUMMER using analytic techniques due to D IRICHLET. In his book “Essai de théorie des nombres”, published in variuous editions between 1798 and 1830, LEGENDRE gave variants of this proof but was not able to fill the gaps completely. See p. 4. C.F. Gauss, Disquisitiones Arithmeticae, Braunschweig 1801. Untersuchungen über die Primzahlen, deren Reste oder Nichtreste gegebene Zahlen sind, art. 107 ff. By extending a technique already used by F ERMAT and EULER , GAUSS proves the quadratic reciprocity law using induction. The actual proof begins in art. 135 of the Disquisitiones. See p. 7 and p. 85. C.F. Gauss, Disquisitiones Arithmeticae, Braunschweig 1801. Zweiter Beweis des Fundamentalsatzes und der übrigen auf die Reste 1, C2, 2 sich beziehenden Sätze, art. 262 ff. Using his theory of binary quadratic forms, in particular the theory of ambiguous forms, GAUSS gives a second proof of the quadratic reciprocity law. The technique involved was extended by KUMMER in his proof of the `-th power reciprocity law in cyclotomic number fields, and was subsequently generalized into a substantial piece of class field theory. See p. 63. C.F. Gauss, Theorematis arithmetici demonstratio nova, Comment. Soc. regiae sci. Göttingen XVI (1808), 69; Werke II (1863), p. 1–8 GAUSS’s third proof using GAUSS’s Lemma became extremely popular, and was used for many (if not the most) proofs published afterwards. See p. 15 and p. 90.

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15 Proofs of the Quadratic Reciprocity Law C.F. Gauss, Summatio serierum quarundam singularium, Comment. Soc. regiae sci. Göttingen 1811; Werke II (1863), p. 9–45 GAUSS’s fourth proof is based on the determination of the sign of quadratic Gauss sums, and is perhaps the deepest of the proofs given by G AUSS. See p. 48. C.F. Gauss, Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et amplicationes novae, 1818; Werke II (1863), 47–64, in particular p. 51 This proof is, like the third proof [4], based on G AUSS’s Lemma. See p. 17 and p. 91. C.F. Gauss, Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et amplicationes novae, 1818; Werke II (1863), 47–64, in particular p. 55 GAUSS’s sixth proof uses quadratic Gauss sums, but here G AUSS manages to do without the complicated determination of the sign of Gauss sums. Variations of this proof were published later by CAUCHY Cauchy-G6, JACOBI [9] and EISENSTEIN . See p. 52. A.L. Cauchy, Sur la théorie des nombres, Bull. de Férussac 12 (1829), 205–221; Œuvres S. 2, II, 88–107. C AUCHY presents a variant of GAUSS’s sixth proof. See p. 55. C.G. Jacobi, see A.M. Legendre, Essai sur la théorie des nombres, third edition Paris 1830; German transl. (Zahlentheorie) of the 3rd. ed., Leipzig 1886; Quatrième édition conforme à la troisième, nouveau tirage corrigé, Paris 1955. LEGENDRE publishes a variant of GAUSS’s sixth proof which he had received from JACOBI . See p. 55. L. Dirichlet, Über eine neue Anwendung bestimmter Integrale auf die Summation endlicher oder unendlicher Reihen, Abh. Preuss. Akad. Wiss. 1835, 649–664; Werke I, 237–256; French transl.: Sur l’usage des intégrales définies dans la sommation des séries finies ou infinies, J. Reine Angew. Math. 17 (1837), 57–67; Werke I, 257–270 DIRICHLET determines the sign of the quadratic Gauss sum using analytic methods and presents a variant of GAUSS’s fourth proof. See p. 113 V.A. Lebesgue, Recherches sur les nombres, J. math. pures appl. 3 (1838), 113–144 V.A. L EBESGUE (see Théorème II on p. 134 of this article) gives a proof of the quadratic reciprocity law based on counting the number of solutions of the congruence x12 C : : : C xq2  1 mod p. His idea was later resurrected by ELY , L EMMERMEYER and CASTRYCK , and also showed up as Exercise 40 in G. SCHEJA and U. STORCH , [Lehrbuch der Algebra, Teil 2, Teubner 1988]; see U. STORCH , Pythagoras and Diophantus, Resonance, July 2009, 691–703. V.A. Lebesgue, Recherches sur les nombres, J. math. pures appl. 3 (1838), 113–144 The second proof given in this article is essentially GAUSS’s third proof; see the corollary on p. 142 of L EBESGUE’s article. Th. Schönemann, Theorie der symmetrischen Functionen der Wurzeln einer Gleichung. Allgemeine Sätze über Congruenzen nebst einigen Anwendungen derselben, J. Reine Angew. Math. 19 (1839), 289–308 SCHÖNEMANN studies the roots of the quadratic period equation; he also gives applications of the same idea to higher power residues. In this article, SCHÖNEMANN also stated and proved what later became known as SCHOLZ’s reciprocity law. A. L. Cauchy, Méthode simple et nouvelle pour la détermination complete de sommes alternées, formées avec les racines primitives des equations binômes, C. R. Acad. Sci. Paris 10 (1840), 560–572; J. Math. Pures Appl. 5 (1840), 154–168; Œuvres I 5 (1885), 152–166 C AUCHY uses analytic tools for determining the sign of the quadratic Gauss sum. He very briefly sketches (p. 160–161) how to derive the quadratic reciprocity law from these results. See p. 55. G. Eisenstein, Neuer und elementarer Beweis des Legendre’schen Reciprocitäts-Gesetzes, J. Reine Angew. Math. 27 (1844), 322–329; Math. Werke I, 100–107 E PISENSTEIN’S proof is based on manipulations of what we now call multiple Jacobi sums .˛1 =p/    .˛ =p/. See p. 56.

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[28]

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G. Eisenstein, La loi de réciprocité tirée des formules de Mr. Gauss, sans avoir déterminée préalablement la signe du radical, J. Reine Angew. Math. 28 (1844), 41–43; Math. Werke I, 114–116 EISENSTEIN refers to G AUSS’s fourth proof and D IRICHLET’s determination of the sign of the quadratic Gauss sum, and writes that G AUSS’s considerations necessarily require the determination of this sign. It almost seems that, at this point, he was not yet familiar with G AUSS’s sixth proof [7] although he mentioned that Gauss had given six different proofs in [15]. See p. 55. G. Eisenstein, Geometrischer Beweis des Fundamentaltheorems für die quadratischen Reste, J. Reine Angew. Math. 28 (1844), 246–248; Math. Werke I, 164–166; Engl. Transl. Quart. J. Math. 1 (1857), 186–191, or A. Cayley: Coll. Math. Papers III, 39–43 This is E ISENSTEIN ’s beautiful geometric proof of the quadratic reciprocity law. See p. 19 and p. 93. G. Eisenstein, Application de l’algèbre à l’arithmétique transcendante, J. Reine Angew. Math. 29 (1845), 177–184; Math. Werke I, 291–298 This note contains E ISENSTEIN’s proof of the quadratic reciprocity law using the sine function, along with an analogous proof of the biquadratic reciprocity law. See p. 41. E.E. Kummer, Über die Divisoren gewisser Formen der Zahlen, welche aus der Theorie der Kreistheilung entstehen, J. Reine Angew. Math. 30 (1846), 107–116; Coll. Papers I, 103–116 KUMMER attaches certain forms to subfields of the field of p-th roots of unity and characterizes their prime divisors. K UMMER does not explicitly state the quadratic reciprocity law, but it follows from his results in the special case where e D 2. J. Liouville, Sur la loi de réciprocité dans la théorie des résidus quadratiques, J. math. pure appl. (I), 12 (1847), 95–96; see also C. R. Acad. Sci. Paris 24 (1847), 577–578 LIOUVILLE gives a very short cyclotomic proof of the quadratic reciprocity law. See p. 60. V.A. Lebesgue, Démonstration nouvelle élémentaire de la loi de réciprocité de Legendre, par. M. Eisenstein, précédée et suivie de remarques sur d’autres démonstrations, que peuvent être tirées du même principe, J. math. pures appl. 12 (1847), 457–473 V.A. Lebesgue, see [21]. V.A. Lebesgue, see [21]. V.A. Lebesgue, see [21]. L EBESGUE explains that the proofs given by JACOBI , CAUCHY and EISENSTEIN are minor modifications of GAUSS’s sixth proof. E ISENSTEIN makes essentially the same remark in [25]. LEBESGUE presents a slight modification of E ISENSTEIN ’s proof, and observes that LIOUVILLE’s proof from [20], which he also presents, is shorter, and is closely related to E ISENSTEIN ’s proof [18] using sines. Next LEBESGUE gives E ISENSTEIN’s proof [15] using multiple Jacobi sums, as well as his own proof [11]. He fails to mention the connection between these last two proofs, however. G. Eisenstein, Beiträge zur Theorie der elliptischen Funktionen VI. Genaue Untersuchung der unendlichen Doppelprodukte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, J. Reine Angew. Math. 35 (1847), 153–274; Math. Werke I, 457–478 EISENSTEIN’s new proof of the quadratic reciprocity law canQbe found on pp. 464–465 (Werke I). It is based on manipulations of the function .x/ D .1  mx /, and is basically a minor variation of his proof using the sine function. M. Schaar, Nouvelle démonstration de la loi de réciprocité pour les residus quadratiques, Bulletin de l’Academie de Belgique 14 (1847), 79–83 G. Plana, Memoria su una nuova soluzione algebrica dell’equazione a due termini x n 1 D 0; essendo n un numero primo, Mem. dell Acad. Sci. Torino (2) 11 (1851), 413–468 PLANA ’s proof is based on GAUSS’s sixth proof. PLANA calls it a simplification of JACOBI ’s proof. M. Schaar, Mémoire sur la théorie des residus quadratiques, Acad. Roy. Sci. Lettres Beaux Arts Belgique 24 (1852), 14 pp

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15 Proofs of the Quadratic Reciprocity Law This proof by SCHAAR has its origin in G AUSS’s fourth proof, based on the evaluation of quadratic Gauss sums. A. Genocchi, Note sur la théorie des residus quadratiques, Mém. cour. et mém. des savants étrangers Acad. Roy Sci. Lettres Belgique 25 (1851/53), 54 pp A. Genocchi, see [29]. A. Genocchi, see [29]. GENOCCHI determines the sign of quadratic Gauss sums using analytic tools and deduces the quadratic reciprocity law along the lines of G AUSS’s fourth proof. In addition, he gives two variations of the proof of Liouville and a proof similar to EISENSTEIN’s proof using the sine function, which he was unaware of, as he remarks in his article from 1885. See p. 20 and p. 98. L. Dirichlet, Über den ersten der von Gauss gegebenen Beweise des Reciprocitätsgesetzes in der Theorie der quadratischen Reste, J. Reine Angew. Math. 47 (1854), 139–150; Werke II, 121–138 DIRICHLET presents a simplified version of GAUSS’s first proof. A. Genocchi, Note sur les nombres complexes, J. Math. Pures Appl. 19 (1854), 281–288 GENOCCHI uses the arithmetic of cyclotomicields for proving FERMAT’s Theorem and WILSON’s theorem, and then presents a variation of L IOUVILLE’s proof using cyclotomy. M. Schaar, Recherches sur la théorie des residus quadratiques, Acad. Roy. Sci. Lettres Beaux Arts Belgique 25 (1854), 20 pp SCHAAR determines the sign of quadratic Gauss sums by analytic means and derives the quadratic reciprocity law. V.A. Lebesgue, Note sur les congruences, C. R. Acad. Sci. Paris 51 (1860), 9–13 LEBESGUE proves the quadratic reciprocity law by investigating the roots of the quadratic period equation. J.J. Sylvester, Sur la fonction E.x/, C. R. Acad. Sci. Paris 50 (1860), 732–734; Mathematical Papers vol. II, art. 30, 179–180. SYLVESTER gives a new proof of Eisenstein’s claim on lattice points inside a rectangle, from which the quadratic reciprocity law follows. E.E. Kummer, Zwei neue Beweise der allgemeinen Reziprozitätsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist, Berliner Akad. Abh. 1861; J. Reine Angew. Math. 100 (1887), 10–50; Coll. Papers I, 842–882 E.E. Kummer, see [37]. KUMMER ’s proofs use the theory of binary quadratic forms and the existence of auxiliary primes, whose existence follow from DIRICHLET’s techniques. Although he does not mention it, KUMMER ’s observations close the gap in L EGENDRE’s incomplete proof. See p. 65 and p. 68. L. Dirichlet, R. Dedekind, Vorlesungen über Zahlentheorie; Braunschweig 1863 D EDEKIND ’s proof can be found in Suppl. X, § 154. It is a variation of G AUSS’s second proof based on genus theory of binary quadratic forms. C.F. Gauss, 7th Proof, Werke II (1863), p. 233 C.F. Gauss, 8th Proof, Werke II (1863), p. 234 GAUSS’s 7th and 8th proof are based on the quadratic period equation. Proofs with similar approaches have meanwhile been given by S CHÖNEMANN [13] and LEBESGUE [35]. M. Jenkins, Proof of an Arithmetical Theorem leading, by means of Gauss’ fourth demonstration of Legendre’s law of reciprocity, to the extension of that law, Proc. London Math. Soc. 2 (1867), 29–32 JENKINS generalizes G AUSS’s Lemma to composite integers, and then proves the quadratic reciprocity law for JACOBI symbols by using GAUSS’s evaluation of quadratic Gauss sums. E. Mathieu, Mémoire sur la théorie des résidus biquadratiques, J. de Math. Pures Appl. (2) 12 (1867), 377–438 M ATHIEU ’s proof is based on the quadratic period equation, and is closely related to GAUSS’s 7th and 8th proofs, which had not yet been published at that time.

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1/ C. von Staudt, Ueber die Functionen Y und Z, welche der Gleichung 4.xx1 D Y 2  pZ 2 Genüge leisten, wo p eine Primzahl der Form 4k ˙ 1 ist, J. Reine Angew. Math. 67 (1867), 205–217 VON STAUDT considers the polynomials Y; Z 2 ZŒX satisfying the equation Z 2  p 1 p  Y 2 D 4 XX1 for some prime number p D 2q C 1 (their existence was proved by p G AUSS in his Disquisitiones), and deduces the reciprocity law . m /. m / D .1/.p1/.m1/=4 p q q1 q1 by writing Y D 2x C C1 x C : : : C Cq , Z D D1 x C : : : C Dq , and deriving certain congruences from the fact that Cm and Dm are integral. F.W.A. Heime, Untersuchungen, besonders in Bezug auf relative Primzahlen, primitive und sekundäre Wurzeln, quadratische Reste und Nichtreste; nebst Berechnung von primitiven Wurzeln von allen Primzahlen zwischen 1 und 1000, Schulprogramm, Berlin 1868 H EIME gives a proof using Gauss’s Lemma. W.J. Bouniakowski, Sur un théorème relatif à la théorie des résidus et son application à la démonstration de la loi de réciprocité de deux nombres premiers, Bull. Acad. St. Pétersbourg 14 (1869), 432–447 B OUNIAKOWSKI presents a proof similar to G AUSS’s third proof; see p. 28 and p. 95. M.A. Stern, Über einen einfachen Beweis des quadratischen Reciprocitätsgesetzes und einige damit zusammenhängende Sätze, Gött. Nachr (1870), 237–253 STERN proves the quadratic reciprocity law using a modification of G AUSS’s Lemma. BAUMGART’s claim that this proof has a gap (see p. 22) was shown to be incorrect by GEGENBAUER [98]. See also p. 94. Ch. Zeller, Beweis des Reciprocitätsgesetzes für die quadratischen Reste, Berl. Monatsber. (1872), 846–847 ZELLER first revealed his proof in a letter to D EDEKIND dated July 8, 1872. As D EDEKIND writes in [146], the version ZELLER published a few months later is slightly different from the one given in his letter. ZELLER was a mathematical amateur; he was priest in Weiler near Schorndorf (near Stuttgart). See p. 24 and p. 94. ZELLER ’s proof was presented to the Berlin Academy by K RONECKER, and accordingly shows up in Kronecker’s Werke V, 445–448. G. Zolotarev, Nouvelle démonstration de la loi de réciprocité de Legendre, Nouv. Ann. Math (2), 11 (1872), 354–362 Z OLOTAREV’s proof is based on the insight that GAUSS’s Lemma can be formulated as a result on permutations. His proof is among those that are very well known. E. Schering, Verallgemeinerung des Gauss’schen Criteriums für den quadratischen Restcharakter einer Zahl in Bezug auf eine andere, Berl. Ber. (1876), 330–331; Werke I, 285–286 SCHERING presents a variation of GAUSS’s third proof. See p. 98. L. Kronecker, Ueber das Reciprocitätsgesetz, Monatsber. Berlin (1876), 331–341; Werke II, 11–23; French transl. Sur la loi de reciprocité, Bull. Sci. Math. 4 (1880), 182–192; Werke II, 25–36 K RONECKER remarks that he has independently found SCHERING’s generalization of G AUSS’s Lemma to composite integers by investigatingQEISENSTEIN ’s proof using the sine function. Then he derives the main equation . mn / D . mh  kn / in an independent way, which actually uses GAUSS’s main lemma from his first proof. P. Mansion, On the law of reciprocity of quadratic residues, Mess. Math. (2) 5 (1876), 140–143 Mansion gives Zeller’s proof of the quadratic reciprocity law, but deduces Euler’s criterion without using primitive roots. R. Dedekind, Sur la théorie des nombres entiers algébriques, Bull. Sci. Math. Astr. XI (1877); Engl. Transl. by John Stillwell: Theory of algebraic integers, Cambridge Univ. Press. 1996, in particular § 27. DEDEKIND gives a very conceptual proof of the quadratic reciprocity law by comparing the splitting of primes in quadratic and cyclotomic fields. p

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R. Dedekind, Schreiben an Herrn Borchardt über die Theorie der elliptischen Modulfunktionen, J. Reine Angew. Math. 83 (1877), 265–292; Ges. Werke I, 174–201 In a letter to BORCHARDT, D EDEKIND derives the quadratic reciprocity law from a reciprocity law concerning Dedekind sums (see. Werke I, p. 195). A.E. Pellet, Sur la décomposition d’une fonction entière en facteurs irréducibles suivant un module permier, Comptes Rendus Paris 86 (1878), 1071–1072 PELLET proves the quadratic reciprocity law using discriminants; this idea was later taken up by STICKELBERGER , VORONOI, M IRIMANOFF & HENSEL [135], LASKER [157], SWAN [208], BARRUCAND & L AUBIE [246] and ZAHIDI [272]. T. Pépin, Mémoire sur les lois de réciprocité relatives aux résidus des puissances, Atti della Accademia Pontificia dei Nuovi Lincei Roma 31 (1878), 40–149 J. Sochocki, Wyznaczenie Stalych Mnoznikow we wzorach dla linijnei transformacyi funkcyi . Summy Gauss’a i prawo wzajemnosci symbolow Legendre’a (Determination of the constant factors in the transformation formulas of theta functions. Gauss sums and the reciprocity law for Legendre symbols), Par. Denkschrift (1878) SOCHOCKI determines a certain constant in a transformation formula for theta functions that had already been determined by H ERMITE [Sur quelques formules relatives à la transformation des fonctions elliptiques (Liouville J. 1858)] in a way different from HERMITE. He then derives formulas due to GAUSS and CAUCHY in a similar manner as H ERMITE, and develops properties of LEGENDRE and JACOBI symbols. E. Schering, Neuer Beweis des Reciprocitäts-Satzes für die quadratischen Reste, Gött. Nachr. (1879), 217–224; Werke I, 331–336; French transl.: Nouvelle démonstration de la loi de réciprocité dans la théorie des résidus quadratique, C. R. Acad. Sci. Paris 88 (1879), 1073–1075; Werke I, 337–340 SCHERING ’s proof is discussed on p. 31 and p. 90. J. Petersen, A new proof of the theorem of reciprocity, Amer. J. Math. pure and appl. 2 (1879), 285–286 PETERSEN’s proof is similar to Z ELLER’s. See p. 33 and p. 94. A. Genocchi, Sur la loi de réciprocité de Legendre étendue aux nombres non premiers, C. R. Acad. Sci. Paris 90 (1880), 300–302; reprinted in Bull. Bibl. Storia Sci. Mat. Fis. 18 (1885), 235–237 GENOCCHI recalls his proofs of the quadratic reciprocity law given in [29] and [33], and then uses SCHERING ’s generalization of GAUSS’s Lemma to composite integers for giving another proof. A. Genocchi, Sur quelques théorèmes qui peuvent conduire à la loi de réciprocité de Legendre, Bull. bibliogr. storia sci. mat. fis. 18 (1885), 238–243 GENOCCHI gives a cyclotomic proof of the quadratic reciprocity law similar to LIOUVILLE’s. L. Kronecker, Ueber den vierten Gauss’schen Beweis des Reciprocitätsgesetzes für die quadratischen Reste, Monatsber. Berlin (1880), 686–698; Werke IV, 275–294 KRONECKER compares C AUCHY’s determination of the quadratic Gauss sum with that given by D IRICHLET, and deduces the quadratic reciprocity law. L. Kronecker, Ueber die Potenzreste gewisser complexer Zahlen, Monatsber. Berlin (1880), 404–407; Werke II, 95–101 K RONECKER proves the quadratic reciprocity law using the quadratic period equation; this boils down to comparing the splitting of primes in quadratic and cyclotomic fields. A. Voigt, Abkürzung des dritten Gauss’schen Reciprocitätsbeweises, Z. Math. Phys. 26 (1881), 134 VOIGT, in a letter to J. Thomae written in 1877, presented a variant of GAUSS’s third proof of the quadratic reciprocity law. THOMAE now publishes VOIGT’s proof since the author had passed away in the meantime. See p. 34 and p. 90. A.E. Pellet, Sur les résidus cubiques et biquadratiques suivant un module premier, Bull. Soc. Math. France 10 (1882), 157–162 PELLET presents a version of M ATHIEU’s proof [43] and then applies the same reasoning to cubic and quartic residues.

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E. Busche, Ueber eine Beweismethode in der Zahlentheorie und einige Anwendungen derselben, insbesondere auf das Reziprozitätsgesetz in der Theorie der quadratischen Reste, Diss. Göttingen 1883 BUSCHE presents a variation of GAUSS’s third proof; see p. 35 and p. 97. L. Gegenbauer, Über das quadratische Reciprocitätsgesetz, Wiener Ber. 90 (1884), 1026– 1035 L. Gegenbauer, see [67]. L. Gegenbauer, see [67]. G EGENBAUER first proves the quadratic reciprocity law using Gauss’s Lemma, and then presents the proofs by K RONECKER [70] and SCHERING [58]. L. Kronecker, Beweis des Reciprocitätsgesetzes für die quadratischen Reste, Monatsber. Berlin (1884), 519–539; Werke II, 498–522; an excerpt was published under the same title in J. Reine Angew. Math. 96, 348–351; Werke II, 523–526 K RONECKER takes up his proof given in [51] again. H. Bork, Untersuchungen über das Verhalten zweier Primzahlen in Bezug auf ihren quadratischen Restcharakter, Diss. Halle, Programm Askem. Gymn. Berlin 1885, 21 pp; BORK first presents EISENSTEIN ’s geometric proof of the quadratic reciprocity law, and then derives several very special cases of the reciprocity law using similar arguments. A. Genocchi, Remarques sur une demonstration de la loi de réciprocité, C. R. Acad. Sci. Paris 101 (1885), 425–427 GENOCCHI points out that ZELLER’s proof essentially coincides with the one he has given in [29]. E. Schering, Zum dritten Gauss’schen Beweis des Reciprocitätssatzes für die quadratischen Reste, Berl. Ber. (1885), 113–117; Werke II, 103–106 E. Schering, see [72]. S CHERING proves the quadratic reciprocity law using GAUSS’s Lemma and compares his proof with KRONECKER ’s. Then he presents a second proof using results he has obtained in earlier publications. L. Kronecker, Ueber den dritten Gauss’schen Beweis des Reciprocitätsgesetzes für die quadratischen Reste, Monatsber. Berlin 1884, 645–649. This article was reprinted as Der dritte Gauss’sche Beweis des Reciprocitätsgesetzes für die quadratischen Reste, in vereinfachter Darstellung, J. Reine Angew. Math. 97 (1885), 93–94; Werke II, 533–536. K RONECKER further simplifies his proof of the quadratic reciprocity law given in [51] and [70]. L. Kronecker, Zum dritten Gauss’schen Beweis des Reciprocitätsgesetzes für die quadratischen Reste (Bemerkungen zu Herrn Ernst Schering’s Mitteilung), Monatsber. Berlin (1885), 117–118; Werke II, 537–540 KRONECKER gives yet another simplification of his proof in [51, 70, 74]. L. Kronecker, Die absolut kleinsten Reste reeller Grössen, Monatsber. Berlin (1885), 383– 386; Werke IIIa, 111–136 L. Kronecker, see [76]. L. Kronecker, see [76]. In this article, KRONECKER gives three different proofs of the identity sgn

Y  hn  Y  km  1 R R D .1/ 4 .m1/.n1/ ; m n h k

where the products are over half systems mod m and mod n, and from which the quadratic reciprocity law follows via GAUSS’s Lemma, which can be given the form sgn

Y  hn   n  R D : m m h

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15 Proofs of the Quadratic Reciprocity Law The first method is a modification of G AUSS’s third proof [70, p. 522], the third essentially coincides with that of ZELLER [48]. K RONECKER also studies GENOCCHI ’s proof in [60]. W. Bock, Ueber eine neue zahlentheoretische Funktion, Hamb. Mitt. 6 (1886), 187–194; BOCK presents another variation of G AUSS’s third proof. S. Eichenberg, Über das quadratische Reciprocitätsgesetz und einige quadratische Zerfällungen der Primzahlen, Diss. Gött., 1886 S. Eichenberg, see [80]. S. Eichenberg, see [80]. EICHENBERG first presents a minor modification of SCHERING ’s proof [50], and then adds two more proofs that use ZELLER’s Lemma. J. Hermes, Beweis des quadratischen Reciprocitätsgesetzes durch Umkehrung, Arch. Math. Phys. (2), 5 (1887), 190–198 H ERMES defines a symbol .s W t / for which the reciprocity law holds, and then proves that it coincides with the Jacobi symbol. He has to use GAUSS’s Lemma on the existence of auxiliary primes from GAUSS’s first proof [2]. M. Lerch, Modification de la troisième démonstration donnée par Gauss de la loi de reciprocité de Legendre, J. Sciencias Matem. Astron. 8 (1887), 137–146 LERCH gives a variation of G AUSS’s third proof. E. Busche, Ueber grösste Ganze, J. Reine Angew. Math. 103 (1888), 118–125 BUSCHE proves the quadratic reciprocity law using G AUSS’s Lemma, and mentions possible applications to the biquadratic reciprocity law. J. Hacks, Schering’s Beweis des Reciprocitäts-Satzes für die quadratischen Reste dargestellt mit Hülfe des Zeichens Œx, Acta Math. 12 (1889), 109–111 HACKS presents a modification of SCHERING’s proof of the quadratic reciprocity law. L. Kronecker, Beweis des Reciprocitätsgesetzes für die quadratischen Reste, J. Reine Angew. Math. 104 (1889), 348–351; Werke III, 137–144 KRONECKER modifies his proof given in [70] and compares it with proofs by GENOCCHI , SCHERING and H ACKS. A. Tafelmacher, Zu dem dritten Gauss’schen Beweise des Reciprocitäts-Satzes für die quadratischen Reste gehörende Untersuchungen, Diss. Göttingen 1889, Pr. Gymn. Osnabrück 1890, 1–24; A. Tafelmacher, see [88]. A. Tafelmacher, see [88]. TAFELMACHER presents three proofs of the quadratic reciprocity law by modifying proofs by STERN [47] and SCHERING (1885). P.q1/=2 , J. Reine Angew. Math. 106 (1890), 65–80 E. Busche, Ueber die Function xD1 Œ px q BUSCHE modifies his own proof of the quadratic reciprocity law given in [66]; apparently he was not content with BAUMGART’s presentation of his proof. F. Franklin, A proof of the theorem of reciprocity for quadratic residues, Mess. Math. (2), 19 (1890), 176–177 FRANKLIN gives a proof of the quadratic reciprocity law similar to GAUSS’s third proof. L. Kronecker, Ueber die Dirichlet’sche Methode der Wertbestimmung der Gaussschen Reihen, Festschr. Hamb. Mitt. 2 (1890), 32–36; Werke IV, 301–308 K RONECKER discusses DIRICHLET’s determination of quadratic Gauss sums and deduces the quadratic reciprocity law. E. Lucas, Nouvelle démonstration de la loi de réciprocité, Assoc. Franç. Limoges 191 (1890), 147; see also Sur la loi de réciprocité des résidus quadratiques, St. Petersbourg, Mélanges math. et astr. 8, 65–66 LUCAS proves the quadratic reciprocity law for JACOBI symbols using G AUSS’s Lemma. T. Pépin, Nouvelle démonstration de la loi de réciprocité de Legendre, Rom. Acc. Pont. d. Nuovi Lincei 43 (1890), 192–198 PÉPIN proves the quadratic reciprocity law using GAUSS’s genus theory.

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J. C. Fields, A simple statement of proof of reciprocal theorem, American J. Math. 13 (1891), 189–190 FIELDS proves the quadratic reciprocity law using G AUSS’s Lemma. He generalizes p1 q1 G AUSS’s congruence  C   2 2 mod 2 to integers p; q with gcd.p; q/ D r, in q1 which case he finds  C   p1 C r1 mod 2. 2 2 2 L. Gegenbauer, Note über das Legendre-Jacobi’sche Symbol, Wiener Ber. 100 (1891), 855–864 GEGENBAUER proves the quadratic reciprocity law using G AUSS’s Lemma, and deduces the second supplementary law from the general law. L. Gegenbauer, Beweis des quadratischen Reciprocitätsgesetzes, Monatsh. f. Math. 4 (1893), 190–192 GEGENBAUER (1893) corrects a remark by Baumgart concerning STERN’s proof of the quadratic reciprocity law. L. Gegenbauer, Arithmetische Untersuchungen, Denkschrift Wien, 1893 L. Gegenbauer, see [99] L. Gegenbauer, see [99] G EGENBAUER presents modifications of the proofs by ZELLER and PETERSEN, as well as a proof of his own. G. Heinitz, Eine neue Bestimmung des quadratischen Restcharakters, Wiss. Beil. z. Pr. Realsch. Seesen a. Harz 694, Göttingen 1893, 45 pp HEINITZ’s manuscript was not available to me. H. Schmidt, Drei neue Beweise des Reciprocitätssatzes in der Theorie der quadratischen Reste, J. Reine Angew. Math. 111 (1893), 107–120 H. Schmidt, see [103]. H. Schmidt, see [103]. SCHMIDT gives two proofs of the quadratic reciprocity law using GAUSS’s Lemma and a proof by induction. L. Gegenbauer, Einige Bemerkungen zum quadratischen Reciprocitätsgesetz, Wiener Ber. 103 (1894), 285–294 F. Hasenöhrl, Über das quadratische Reziprozitätsgesetz, Seminararbeit 1894; cf. Anzeiger K. Akad. Wien 31 (1894), 74–76 HASENÖHRL proves the quadratic reciprocity law using G AUSS’s Lemma. A. S. Bang, Nyt Bevis for Reciprocitetsaetninger, Nyt. Tidss. for Math. V B (1894), 92–96 F. Mertens, Ueber den quadratischen Reciprocitätssatz und die Summen von Gauss, Wiener Ber. 103 (1894), 1005–1022 F. Mertens, see [109] MERTENS gives two proofs of the quadratic reciprocity law, one similar to K RONECKER ’s proof via signs, and another one based on cyclotomy, or, to be more precise, on Fekete polynomials. He uses the same method for determining the sign of the quadratic Gauss sum. E. Busche, Beweis des quadratischen Reciprocitätsgesetzes, Hamburger Mitt. 3 (1896), 233–234 BUSCHE proves using a geometric interpretation of G AUSS’s Lemma. E. Busche, Über eine allgemeine Anzahlbeziehung und einige Anwendungen davon auf die Zahlentheorie, Hamburger Mitt. 3 (1896), 333–346 P P p1 q1 BUSCHE proves G AUSS’s identity Œ px  C Œ qx  D 2 2 , from which the quadratic q p reciprocity law follows. E.J. Lange, Ein elementarer Beweis des Reciprocitätsgesetzes, Ber. Verh. Sächs. Akad. Wiss. Leipzig 48 (1896), 629–633 LANGE gives a proof using a geometric interpretation of G AUSS’s Lemma similar to EISENSTEIN’s. C. de la Vallée Poussin, Recherches arithmétiques sur la composition des formes binaires quadratiques, Mém. Acad. Belgique 53 (1895/86). no. 3, 59 pp.; Coll. Works vol I, Acad. Royale Belgique (2000)

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15 Proofs of the Quadratic Reciprocity Law According to M ANSION’s “rapport”, this is an exposition of GAUSS’s genus theory and his second proof of the quadratic reciprocity law. P. Mansion, Rapport, Belg. Bull. (3) 30 (1896), 189–193 This is a review of DE LA VALLÉE POUSSIN ’s work on composition of binary quadratic forms and his exposition of GAUSS’s second proof of the quadratic reciprocity law. E.J. Lange, Ein elementarer Beweis des Reciprocitätsgesetzes, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl. 49 (1897), 607–610 E.J. Lange, see [115] LANGE gives two variations of a modification of his first proof in [113]; this time the geometric interpretation of G AUSS’s Lemma is replaced by arithmetic calculations. D. Hilbert, Die Theorie der algebraischen Zahlen (Zahlbericht), Jahresber. DMV 4 (1897), 175–546 D. Hilbert, see [117]. In § 69, HILBERT proves the quadratic reciprocity law using genus theory, which he based on the theory of his norm residue symbol. In § 122, H ILBERT derives the quadratic reciprocity law from comparing the splitting of primes in quadratic and cyclotomic extensions. W. P. Alexejevsky, Über das Reciprocitätsgesetz der Primzahlen (Russ.), Samml. Mitt. Math. Ges. Charkov (2) 6 (1898) 200–202 ALEXEJEVSKY gives a proof based on ideas by E ISENSTEIN , SCHERING and K RONECKER . T. Pépin, Dissertation sur deux démonstrations du théorème de réciprocité de Legendre, Rom. Acc. Pont. d. Nuovi Lincei 51 (1898), 123–144 T. Pépin, See [120] PÉPIN closes the gap in LEGENDRE’s proof of the quadratic reciprocity law using methods due to GAUSS, and then presents a modification of GAUSS’s fifth proof. J. König, Das Reciprocitätsgesetz in der Theorie der quadratischen Reste, Acta Math. 22 (1899), 181–192 KÖNIG’s proof, a variant of G AUSS’s first proof by induction that does not require G AUSS’s lemma on the existence of auxiliary primes, contains a gap that so far no one was able to close. M. Lerch, Über die Gaußschen Summen (Czech), Casopis 28 (1899), 1–24 L ERCH gives a proof related to G AUSS’s fourth proof. E. Fischer, Ueber Eisenstein’s Beweis des quadratischen Reciprocitätsgesetzes, Monatsh. f. Math. 11 (1900), 176–182 FISCHER interprets EISENSTEIN’s proof in terms of resultants. W. Scheibner, Zur Theorie des Legendre-Jacobi’schen Symbols . mn /, Leipz. Abh. 24 (1899), 369–410 W. Scheibner, see [124]. W. Scheibner, see [124]. W. Scheibner, see [124]. W. Scheibner, see [124]. W. Scheibner, see [124]. W. Scheibner, see [124]. SCHEIBNER extends various proofs of the quadratic reciprocity law for the Legendre symbols to Jacobi symbols (and to not necessarily positive integers). In particular, he presents variations of the proofs by ZELLER (p. 376–377), KRONECKER ’s variation of GAUSS’s third proof (p. 381), G AUSS’s original version, EISENSTEIN (geometric, p. 381; sine, p. 382–383), GAUSS (fourth proof, p. 386, p. 396) E. McClintock, On the nature and use of the functions employed in the recognition of quadratic residues, Trans. Amer. Math. Soc. 3 (1902), 92–109 M C CLINTOCK defines a symbol . kn / by certain properties, and then proves the quadratic reciprocity law for . kn /. Afterwards he shows that . kn / coincides with the LEGENDRE symbol.

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[132] M. Lerch, Über den fünften Gaußschen Beweis des Reziprozitätsgesetzes für die quadratischen Reste, Sep.-Abdr. Sitzungsber. Kgl. Böhm. Ges. d. Wiss. 1903, Prag, 12 pp LERCH presents a variation of GAUSS’s fifth proof. [133] T. Takagi, A simple proof of the law of quadratic reciprocity for quadratic residues, Proc. Phys.-Math. Soc. Japan, Ser. II 2 (1903), 74–78; Collected Papers ?? TAKAGI proves the quadratic reciprocity law in a similar way as Z ELLER. [134] F. Mertens, Über eine Darstellung des Legendreschen Zeichens, Wiener Ber. 113 (1904), 905–910 M ERTENS proves a representation of the Legendre symbols in terms of certain invariants of polynomial rings, and reads off the reciprocity law. His proof resembles EISENSTEIN’s proof via the sine function. / D .1/nh et la loi de réciprocité, J. Reine [135] D. Mirimanoff, K. Hensel, Sur la relation . D p Angew. Math. 129 (1905), 86–87 M IRIMANOFF & HENSEL prove a result on discriminants that implies the quadratic reciprocity law. [136] G. Cornacchia, Sulla congruenza x n C y n  zn .mod p/, Giornale di Mat. (2) 47 (1909), 219–268 CORNACCHIA gives a cyclotomic proof of the quadratic reciprocity law. [137] E. Busche, Eine geometrische Veranschaulichung des quadratischen Restcharakters, Hamb. Mitt. 4 (1909), 403–409 BUSCHE presents a geometric interpretation of Z ELLER’s proof. [138] E. Busche, Zur Theorie der Funktion Œx, J. Reine Angew. Math. 136 (1909), 39–57 [139] E. Busche, see [138]. BUSCHE presents a variation of GAUSS’s third proof (see p. 46), and then gives a second derivation (also by counting lattice points) of the quadratic reciprocity law by proving (p. 56) the equation  a  b  m

n

D"

mn 2

 .1/

m3 n1 2  2

;

where an  bm D " D ˙1. [140] A. Aubry, Exposition élémentaire de la loi de réciprocité dans la théorie des nombres, Ens. Math. 12 (1910), 457–475 [141] A. Aubry, see [140] [142] A. Aubry, see [140] [143] T. Pépin, Théorie des Nombres; Suite et fin, Memorie della Accademia Pontificia dei Nuovi Lincei Roma 29 (1911), 319–339 P [144] K. Petr, Poznámka o Legendre-Jacobiove symbolu . Q / (A remark on the Legendre-Jacobi Symbol .P =Q/) (Czech), Casopis 40 (1911), 162–165 PETR proves the quadratic reciprocity law using ideas from M ERTENS [134]. [145] H.C. Pocklington, The determination of the exponent to which a number belongs, the practical solution of certain congruences, and the law of quadratic reciprocity, Math. Proc. Cambr. Phil. Soc. 16, (1911), 1–5 POCKLINGTON gives an arithmetic proof of the Lemma that E ISENSTEIN had proved by geometric means. [146] R. Dedekind, Über den Zellerschen Beweis des quadratischen Reziprozitätsgesetzes, H. Weber Festschrift Leipzig (1912), 23–36; Ges. Math. Werke II, 340–353 [147] R. Dedekind, see [146, p. 349–350]. [148] R. Dedekind, see [146, p. 350–351]. [149] R. Dedekind, see [146, p. 351]. DEDEKIND presents his version of ZELLER ’s original proof, and then discusses three other proofs based on the same source. [150] P.J. Heawood, On a graphical demonstration of the fundamental properties of quadratic residues, London Math. Soc. Proc. (2), 12 (1913), 373–376

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P.J. Heawood, The law of quadratic reciprocity, Math. Gaz. London 23, 198–200 Heawood gives a proof reminiscent of Eisenstein’s proof via lattice points, but uses a representation of integers on a circle instead. J. McDonnell, On quadratic residues, Trans. Amer. Math. Soc. 14(1913), 477–480 Q km  km , where the product is over a M C DONNELL gives a cyclotomic proof based on  k  k half system mod n. G. Frobenius, Über das quadratische Reziprozitätsgesetz I, Sitzungsberichte Berliner Akad. (1914), 335–349; Ges. Abhandl., 628–642 G. Frobenius, see [152]. G. Frobenius, see [152]. G. Frobenius, see [152]. FROBENIUS presents ZOLOTAREV ’s proof of the quadratic reciprocity law, then gives a geometric interpretations of the proofs by ZELLER, and G AUSS’s fifth and third proofs. G. Frobenius, Über das quadratische Reziprozitätsgesetz II, Sitzungsberichte Berliner Akad. (1914), 484–488; Ges. Abhandl., 643–647 FROBENIUS discusses EISENSTEIN ’s geometric proof and compares it with GAUSS’s third proof. E. Lasker, Über eine Eigenschaft der Diskriminante, Sitzungsber. Berl. Math. Ges. 15 (1916), 176–178 LASKER proves VORONOI’s theorem: if f is a polynomial with degree n and discriminant D, and if p is a prime number coprime to D such that f splits into a factors modulo p, then . Dp / D .1/nCa . This easily implies the quadratic reciprocity law. Compare [55]. A. Cerone, Sulle formole di moltiplicazione delle funzioni circolari e teorema di reciprocità pei residui quadratici, Periodico di Mat. 31 (1916), 171–175 C ERONE gives a proof along the lines of EISENSTEIN ’s proof via the sine function. A. Cerone, Sulla moltiplicazione e divisione dell’argomento nelle funzioni circolari, Periodico di Mat. 32 (1917), 49–60 Another proof similar to EISENSTEIN ’s proof via the sine function. A. L. Bartelds, F. Schuh, Elementair bewijs der uitgebreide wederkeeringheidswet van Legendre, Nieuw Arch. Wisk. 12 (1918), 420–438 BARTELDS & SCHUH prove the quadratic reciprocity law for Jacobi symbols along G AUSS’S third proof. T.J. Stieltjes, Sur la loi de réciprocité de Legendre, Œuvres Complètes II (1918), 567–573 S TIELTJES gives a proof of the quadratic reciprocity law similar to E ISENSTEIN’s geometric proof. H. Teege, Über den Legendreschen Beweis des sogenannten Reziprozitätsgesetzes in der Lehre von den quadratischen Resten und seine Vervollständigung durch den Nachweis, dass jede Primzahl von der Form 8n C 1 quadratischer Nichtrest unendlich vieler Primzahlen von der Form 4n C 3 ist, Mitt. Math. Ges. Hamburg 5 (1920), 6–19 TEEGE gives a direct proof of Legendre’s Lemma required for closing the gap in Legendre’s proof. A. Arwin, A contribution to the theory of closed chains, Annals of Math. (2) 25 (1924), 91–117 ARWIN shows that the quadratic reciprocity law may be deduced from his theory of closed chains; his proof is related to GAUSS’s second proof. C.M. Huber, On the prime divisors of the cyclotomic functions, Trans. Amer. Math. Soc. 27 (1925), 43–48 H UBER proves criteria for p to be an e-th power residue modulo q in terms of period equations. His result implies the quadratic reciprocity law; he only gives applications to the cases e D 3 and e D 4, however. H. Teege, Ein Kreisteilungsbeweis für das quadratische Reziprozitätsgesetz, Mitt. Math. Ges. Hamburg 6 (1925), 136–138 TEEGE proves the quadratic reciprocity law using a modification of JACOBI ’s proof using quadratic Gauss sums.

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[165] L. Rédei, Ein neuer Beweis des quadratischen Reciprocitätssatzes, Acta Sci. Math. Szeged 2 (1925), 134–138 [ * ] L. Rédei, Ein neuer Beweis des quadratischen Reciprocitätssatzes, J. Reine Angew. Math. 155 (1926), 103–106 Articles by Hungarian mathematics published abroad first had to published in Hungary, which is why most of RÉDEI ’s articles were published twice. The proof given here works with half systems modulo pq. [166] R. F. Whitehead, A proof of the law of quadratic reciprocity, J. London Math. Soc. 2 (1927), 51–55 WHITEHEAD ’s proof of the quadratic reciprocity law resembles those given by KUMMER [37]. [167] K. Petr, Über die lineare Transformation der Thetafunktionen (Czech.), Rozpravy Ceske Akad. ved. (2) 36, No. 1 (1927), 10 pp PETR derives the quadratic reciprocity law from transformation formulas of theta functions. [168] Th. Skolem, Geschlechter und Reziprozitätsgesetze, Norsk. Mat. Forenings Skrifter (1), 18 (1928), 38 pp SKOLEM gives a new definition of the genus in algebraic number theory, and proves, among others, the quadratic reciprocity law. [169] K. Petr, Poznámka k dukazu zákona reciprocity pro kvadratiské zbytky (Remarque concernant la loi de réciprocité des residus quadratiques, Czech., French summary), Casopis 62 (1934), 228–230 PETR gives a variant of GAUSS’s third proof (similar to Kronecker’s proof with signs) of the quadratic reciprocity law. [170] S. C. van Veen, De wederkkerigsheidswet der kwadraatresten, Math. Tijds. v. Studeerenden 1 (1934), 148–153; FdM 59 - II (1933), 935; VAN V EEN proves the quadratic reciprocity law by counting lattice points. [171] R. Fueter, Zur Theorie der Brandtschen Quaternionenalgebren, Math. Ann. 110 (1935), 650–661 FUETER deduces the quadratic reciprocity law from the theory of quaternion algebras over the field of rationals. [172] A.L. Whiteman, On the law of quadratic reciprocity, Bull. Am. Math. Soc. 41 (1935), 359–360 WHITEMAN gives a proof of the quadratic reciprocity law using GAUSS’s Lemma. [173] N. R. C. Dockeray, The law of quadratic reciprocity, Math. Gaz. 22 (1938), 440–453 DOCKERAY presents EISENSTEIN’s geometric proof of the quadratic reciprocity law and suggests it might be suitable for school teachers and their classes. [174] H. Kapferer, Über eine periodische Funktion von drei ganz rationalen Veränderlichen und deren zahlentheoretische Bedeutung, Monatsh. Math. Phys. 47 (1939), 285–298 K APFERER studies an arithmetical function, and in this connection proves the quadratic reciprocity law (p. 296) by essentially cyclotomic means. [175] A. Scholz, Einführung in die Zahlentheorie, Göschen 5131 1939 SCHOLZ presents a beautiful proof of Euler’s version of the quadratic reciprocity law using GAUSS’s Lemma. [176] K. Dörge, Beweis des Reziprozitätsgesetzes für quadratische Reste, Mathem. Ann. 118 (1942), 310–311 D ÖRGE proves the quadratic reciprocity law in a similar way as FROBENIUS [152]. [177] L. Rédei, Kurze Darstellung des fünften Gauss’schen Beweises für den quadratischen Reziprozitätssatz, Comment. Math. Helv. 16 (1944), 264–265 RÉDEI gives a short exposition of GAUSS’s fifth proof. [178] H. Lewy, Waves on sloping beaches, Bull. Amer. Math. Soc. 52 (1946), 737–775 LEWY investigates water waves and derives the quadratic reciprocity law from his results; the proof is essentially cyclotomic and uses the sine function. [179] K. Petr, On alternating functions in a cyclotomic field (Czech), Rozpravy Il. Tridy Ceske Akad. 56 (1946), 12 pp

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15 Proofs of the Quadratic Reciprocity Law He proves the PETR extends CAUCHY ’s proof in [14] from primes to composite Q k integers. reciprocity law using the “alternating function” ./ D .   k /, where  is a primitive p-th root of unity. F. Furquim de Almeida, Uma nova demostraçao da lei da reciprocidade quadrática (The law of quadratic reciprocity), Bol. Soc. Mat. Sao Paulo 3 (1948), 3–8 FURQUIM DE A LMEIDA proves the quadratic reciprocity law using Vandermonde determinants. Th. Skolem, A property of ternary quadratic forms and its connection with the quadratic reciprocity theorem (Norw.), Norsk Mat. Tidsskr. 30 (1948), 1–10 SKOLEM proves the quadratic reciprocity law using ternary quadratic forms. It is essentially equivalent to GAUSS’s second proof. A. Aigner, A generalization of Gauss’ lemma, Amer. Math. Monthly 57 (1950), 408–410 AIGNER generalizes GAUSS’s Lemma and proves the quadratic reciprocity law. D. Barbilian, Das Reziprozitätsgesetz mit Anwendung auf die Galoissche Theorie (Roum., French and Russian summaries), Acad. Republ. popul. Romane, Bul. Sti. A 2 (1950), 731–736 BARBILIAN presents a version of G AUSS’s sixth proof. J. Delsarte, Une démonstration de la loi de réciprocité quadratique, Œuvres de Jean Delsarte, vol. 2 (1950), 847–850 DELSARTE gives a proof of the quadratic reciprocity law using Vandermonde determinants. L. Rédei, Einfacher Beweis des quadratischen Reziprozitätssatzes, Mathemat. Z. 54 (1951), 25–26 RÉDEI gives a proof of the quadratic reciprocity law similar to GAUSS’s third. It is based on counting nonnegative solutions of the Diophantine equation px C qy D k. H. Brandt, Über das quadratische Reziprozitätsgesetz, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl. 99 (1951), 17 pp B RANDT presents a new formulation of the quadratic reciprocity law and gives a proof similar to GAUSS’s second proof. H. Brandt, Über das quadratische Reziprozitätsgesetz im rationalen Zahlkörper, Math. Nachr. 6 (1951), 125–128 BRANDT uses quadratic Gauss sums as in G AUSS’s sixth proof for proving his version of the quadratic reciprocity law. B. W. Brewer, On the quadratic reciprocity law, Amer. Math. Monthly 58 (1951), 177–179 BREWER presents a proof based on finite fields, borrowing ideas from G AUSS, LEBESGUE and PELLET. H. J. Zassenhaus, The quadratic law of reciprocity and the theory of Galois fields, Proc. Glasgow Math. Assoc. 1 (1952), 64–71 ZASSENHAUS presents G AUSS’s sixth proof using quadratic Gauss sums in finite fields. M. Riesz, Sur le lemme de Zolotareff et sur la loi de réciprocité des restes quadratiques, Math. Scand. 1 (1953), 159–169 RIESZ gives a proof of the quadratic reciprocity law using Z OLOTAREV’s Lemma. A. Fröhlich, On fields of class two, Proc. London Math. Soc. (3) 4 (1954), 235–256 FRÖHLICH deduces the quadratic reciprocity law from his investigation of the Galois groups of abelian extensions of cyclotomic fields. N. C. Ankeny, The law of quadratic reciprocity, Norske Vid. Selsk. Forh., Trondheim 28 (1956), 145–146 A NKENY uses cyclotomic fields to prove the quadratic reciprocity law. D.H. Lehmer, A low energy proof of the reciprocity law, Amer. Math. Monthly 64 (1957), 103–106 LEHMER proves the quadratic reciprocity law using a geometric interpretation of GAUSS’s Lemma. C. Meyer, Ueber einige Anwendungen Dedekindscher Summen, J. Reine Angew. Math 198 (1957), 143–203 C. Meyer, see [194].

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M EYER gives two proofs of the quadratic reciprocity law. The first is based on Dedekind sums, the second on ideas of ZOLOTAREV . [196] L. Holzer, Zahlentheorie I, Teubner, Leipzig, 1958 H OLZER gives a proof similar to GAUSS’s sixth proof using quadratic Gauss sums in finite fields. [197] L. Rédei, Zur Theorie der Polynomideale über kommutativen nullteilerfreien Hauptidealringen, Math. Nachr. 18 (1958), 313–332 RÉDEI explicitly constructs polynomials u; v 2 ZŒX with xm  1 xn  1 u.x/ C v.x/ D 1 x1 x1 and derives the quadratic reciprocity law from this identity. [198] H. Reichardt, Eine Bemerkung zur Theorie des Jacobischen Symbols, Math. Nachr. 19 (1958), 171–175 REICHARDT extends G AUSS’s Lemma to number fields and then presents a variant of GAUSS’s third proof. [199] H.S. Vandiver, M.W. Weaver, Quadratic Reciprocity, Amer. Math. Monthly 65 (1958), 42–47 VANDIVER & WEAVER present a proof of the quadratic reciprocity law based on those by ZELLER and FROBENIUS. [200] L. Carlitz, A note on Gauss’ first proof of the quadratic reciprocity theorem, Proc. Amer. Math. Soc. 11 (1960), 563–565 CARLITZ presents a streamlined version of G AUSS’s first proof. [201] T. Kubota, Reciprocities in Gauss’ and Eisenstein’s number fields, J. Reine Angew. Math. 208 (1961), 35–50 KUBOTA presents analytic proofs of the cubic and quartic reciprocity law, and at the end of his article sketches how his results also imply the quadratic reciprocity law. [202] T. Kubota, Über quadratische Charaktersummen, Nagoya Math. J. 19 (1961), 15–25 K UBOTA uses H ECKE’s approach to quadratic reciprocity laws using Hecke Gauss sums. [203] T. Kubota, Anwendung Jacobischer Thetafunktionen auf die Potenzreste, Nagoya Math. J. 19 (1961), 1–13 KUBOTA presents a proof of the quadratic reciprocity similar to E ISENSTEIN ’s proof using the sine function, and then proves the biquadratic reciprocity law in a similar way. [204] Th. Skolem, Remarks on proofs by cyclotomic formulas of reciprocity laws for power residues, Math. Scand. 9 (1961), 229–242 [205] Th. Skolem, see [204]. S KOLEM proves the quadratic reciprocity law using the quadratic period equation, and gives a second proof using calculations in cyclotomic fields. [206] Th. Skolem, A proof of the quadratic law of reciprocity with proofs of two so-called “Ergänzungssätze”, Norske Vid. Selsk. Forh. Trondheim 34 (1961), 18–24 S KOLEM proves the quadraticQreciprocity law by studying the action of the Frobenius automorphism on the product .1   a /, where  is a primitive p-th root of unity and a runs through a half system modulo p. [207] A. Hausner, On the quadratic reciprocity theorem, Arch. Math. 12 (1961), 182–183 H AUSNER presents GAUSS’s sixth proof using Gauss sums over finite fields, which is similar to what HOLZER did in [196]. [208] R.G. Swan, Factorization of polynomials over finite fields, Pac. J. Math. 12 (1962), 1099– 1106 S WAN generalizes the theorem of S TICKELBERGER and VORONOI and derives the quadratic reciprocity law. [209] L. Koschmieder, Zu Eisensteins transzendentem Beweis des quadratischen Reziprozitätsgesetzes, Comment. Math. Helv. 37 (1962/63), 235–239

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15 Proofs of the Quadratic Reciprocity Law KOSCHMIEDER gives a proof of the quadratic reciprocity law by replacing the sine function in EISENSTEIN ’s proof by the tangent. M. Gerstenhaber, The 152nd proof of the law of quadratic reciprocity, Am. Math. Month. 70 (1963), 397–398 G ERSTENHABER presents a modification of E ISENSTEIN’s proof using the sine function. H. Rademacher, Lectures on elementary number theory, New York-Toronto-London 1964 RADEMACHER proves the quadratic reciprocity law similar to GAUSS’s fourth proof, and uses the finite FOURIER transform in the relevant calculations. A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211; Œuvres III (1980), 1–69 WEIL rewrites HECKE’s proof of the quadratic reciprocity law in his own language. H.D. Kloosterman, The law of quadratic reciprocity, Indag. Math. 27 (1965), 163–164 KLOOSTERMAN simplifies HOLZER ’s proof [196]. S. Chowla, An algebraic proof of the law of quadratic reciprocity, Norske Vid. Selsk. Forh. (Trondheim), 39 (1966), 59 CHOWLA proves the existence of an element ˛ in a finite field with p 2 elements such that ˛ 2 D .1/.p1/=2 p, and derives the quadratic reciprocity law from it. K. Burde, Reziprozitätsgesetze für Gitterpunktsummen, J. Reine Angew. Math. 226 (1967), 159–174 BURDE studies the transformation properties of certain analytic functions using sums of lattice points, and derives the quadratic reciprocity law from these results. His sums of lattice points are related to Dedekind sums. P. Kaplan, Une démonstration géométrique de la loi de réciprocité quadratique, Proc. Japan Acad. 45 (1969), 779–780 K APLAN presents a geometric interpretation of EISENSTEIN’s proof of the quadratic reciprocity law using the sine function. P. Kaplan, Démonstration des lois de réciprocité quadratique et biquadratique, J. Fac. Sci. Tokyo 16 (1969), 115–145 K APLAN presents a variant of L EBESGUE’s proof of the quadratic reciprocity law by counting solutions of quadratic congruences. T. Kubota, On a classical theta-function, Nagoya Math. J. 37 (1970), 183–189 KUBOTA derives the quadratic reciprocity law (in EULER ’s form) from analytic properties of theta functions. J. Tate, Symbols in arithmetic, Proc. Int. Cong. Math. 1970, vol. I, 201–211; Paris 1971 B. J. Birch, K2 of global fields, Institute on Number theory, Proc. Symp. Pure Math. XX, Amer. Math. Soc. (1971), 87–95 After TATE [ * ] had pointed out that the structure of G AUSS’s first proof could be used for computing the K-group K2 Q it became clear that the quadratic reciprocity law could be derived from results in algebraic K-theory. This is what BIRCH does here. I.V. Reshetukha, Generalized sums for characters and their applications to the reciprocity laws (Russ.), Ukrain. Mat. Z. 23 (1971), 270–276 RESHETUKHA studies multiple Gauss sums and uses them to prove the quadratic and the cubic reciprocity laws. S. Agou, Une démonstration de la loi de réciprocité quadratique, Publ. Dép. Math. (Lyon) 9, fasc. 3 (1972), 55–57; see also Agou’s dissertation Polynômes à coéfficients dans un corps fini, Ph.D. diss. Univ. Lyon, 1976 AGOU proves the quadratic reciprocity law using Vandermonde determinants. J. L. Brenner, Zolotarev’s theorem on the Legendre symbol, Pac. J. Math. 45 (1973), 413– 414 BRENNER proves Z OLOTAREV’s Lemma using matrices, and observes that this implies quadratic reciprocity. T. Honda, Invariant differentials and L-functions; Reciprocity law for quadratic fields and elliptic curves over Q, Rend. Sem. Mat. Univ. Padova 49 (1973), 323–325 HONDA proves the quadratic reciprocity law using formal group laws; this is essentially a proof via Gauss sums.

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[224] J. Milnor, D. Husemöller, Symmetric bilinear forms, Springer-Verlag 1973 MILNOR & HUSEMÖLLER present W EIL’s approach to quadratic reciprocity. [225] D. Zagier, Higher dimensional Dedekind sums, Math. Ann. 202 (1973), 149–172; Z AGIER deduces the quadratic reciprocity law from a reciprocity law for Dedekind sums. [226] C. G. Allander, Gauss’s law of reciprocity – a lucid proof, (Swed.), Nordisk. Mat. Tidskr. 22 (1974), 23–25, 40 ALLANDER proves the quadratic rciprocity law using a geometric interpretation of GAUSS’s lemma. [227] B.C. Berndt, R.J. Evans, Least positive residues and the law of quadratic reciprocity, Delta 4 (1974), 65–69 B ERNDT & EVANS prove the quadratic reciprocity law similar to GAUSS’s third proof. [228] F. Hirzebruch, D. Zagier, The Atiyah-Singer Theorem and Elementary Number Theory, Math. Lecture Series 3, Publish & Perish 1974 H IRZEBRUCH & ZAGIER derive the quadratic reciprocity law from the reciprocity law for Dedekind sums. [229] K. Rogers, Legendre’s theorem and quadratic reciprocity, J. Number Theory 6 (1974), 339–344 ROGERS applies deep analytic tools (S ELBERG’s Lemma) for closing the gap in L GENDRE’s proof. [230] B.C. Berndt, A generalization of a theorem of Gauss on sums Involving Œx, Amer. Math. Monthly 82 (1975), 44–51 P P B ERNDT generalizes the identity Œpr=q C Œqr=p D 14 .p  1/.q  1/ due to G AUSS and EISENSTEIN , where r runs through half systems mod q and mod p, respectively, from which the quadratic reciprocity law follows by G AUSS’s Lemma. [231] P. Castaldo, I grafi e la legge di reciprocità dei numeri primi, Archimede 28 (1976), 114– 123 CASTALDO presents a proof of the quadratic reciprocity law based on G AUSS’s Lemma. [232] T.A. Springer, Caracteres quadratiques de groupes abeliens finis et sommes de Gauss, Bull. Soc. Math. France Suppl. Mem. No. 48 (1976), 103–115 S PRINGER presents a Bourbakiesque approach to the quadratic reciprocity law via Gauss sums. The approach is influences by W EIL [212]. [233] K. Burde, Zur Herleitung von Reziprozitätsgesetzen unter Benutzung von endlichen Körpern, J. Reine Angew. Math. 293/294 (1977), 418–427 BURDE presents a cyclotomic proof of the quadratic reciprocity law and then investigates cubic and quartic reciprocity laws. [234] J.B. Friedlander, K.H. Rosen, Refinements of a congruence of Gauss, Elemente d. Math. 32 (1977), 62–64 For coprime positive odd integers k and h let mk .h/ denote the number of elements in the set fhi W 1  j  k1 g with least positive residue mod k larger than k2 . Then the quadratic 2 reciprocity law follows from G AUSS’s Lemma and the congruence mk .h/ C mh .k/ C

h1 k1   0 mod 2: 2 2

FRIEDLANDER & ROSEN prove that this congruence actually holds even mod 4. [235] J. S. Frame, A short proof of quadratic reciprocity, Amer. Math. Monthly 85 (1978), 818–819 FRAME presents a proof similar to KRONECKER ’s proof using the sign function. [236] J. Hurrelbrink, The elements of K2 .ZS /, Manuscripta Math. 24 (1978), 173–177 HURRELBRINK derives the quadratic reciprocity law from a result in algebraic K-theory. [237] L. Auslander, R. Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics?, Bull. Am. Math. Soc., N. S. 1 (1979), 847–897 AUSLANDER & TOLIMIERI present a modern version of GAUSS’s fourth proof.

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[238] K.H. Rosen, A congruence from Dedekind sums, Skr. K. Nor. Vidensk. Selsk. 3 (1979) 1–5 ROSEN proves the congruence mod 4 from [234] using Dedekind sums. [239] W.J. Ryan, Proof of the quadratic reciprocity law in primitive recursive arithmetic, Math. Scand. 45 (1979), 177–197 RYAN proves the quadratic reciprocity law via GAUSS’s Lemma in primitive recursive arithmetic. [240] D.R. Corro, see p. 320–323 in N. Jacobson, Basic Algebra II, Freeman & Co., 1980; 2nd ed. 1989 [241] E. Brown, The first proof of the quadratic reciprocity law, revisited, Amer. Math. Monthly 88 (1981), 257–264 BROWN gives an exposition of G AUSS’s first proof. [242] R. Cuculière, Histoire de la loi de réciprocité quadratique: Gauss et Tate, Study group on ultrametric analysis 1979–1981 36 (1981), 14 pp CUCULIÈRE presents G AUSS’s first proof and compares it with TATE’s proof based on K-theory. [243] D. M. Goldschmidt, Some amazing properties of the function f .x/ D x 2 , Math. Medley 9 (1981), 29–36 GOLDSCHMIDT gives a cyclotomic proof of Euler’s reciprocity law . pn / D . nq / for primes p  q mod 4n. [244] V.G. Kac, Simple Lie groups and the Legendre symbol, in: Algebra, Carbondale 1980, LNM 848 (1981), 110–123 KAC proves the quadratic reciprocity law similar to EISENSTEIN ’s proof via the sine function. [245] S. Barcanescu, On Zolotarev’s proof of the quadratic reciprocity law, Institutul National Pentru Creatie Stiintifica Si Tehnica, Preprint Series in Mathematics no. 15 (1983) BARCANESCU presents a detailed exposition of Zolotarev’s ideas and his proof. [246] P. Barrucand, F. Laubie, Ramification modérée dans les corps de nombres de degré premier, Sém. Théor. Nombres, Bordeaux 1981/82, exposé 13 BARRUCAND & LAUBIE derive the quadratic reciprocity law from a generalization of the theorem of PELLET, S TICKELBERGER and VORONOI. [247] H. Zantema, Global restrictions on ramification in number fields, Manuscr. Math. 43 (1983), 87–106 Z ANTEMA proves a product formula for a cohomological invariant; this product formula is then shown to be equivalent to the quadratic reciprocity law. [248] J. S. Ely, A geometric approach to the quadratic reciprocity law, Comm. Algebra 12 (1984), 1533–1544 ELY uses the unit circle over Fp for deriving the supplementary laws, and derives the 2 quadratic reciprocity law from counting the solutions of x12 C : : : C xm  c mod p. [249] M. Eichler, The quadratic reciprocity law and the elementary theta function, Glasgow J. Math. 27 (1985), 19–30 E ICHLER uses HECKE’s approach to the quadratic reciprocity law and proves a generalization valid in number fields. [250] P. Gérardin, Groupes quadratiques et applications arithmétiques, Sémin. Théor. Nombres Paris 1984–85 (1986), 79–95 G ÉRARDIN recasts W EIL’s proof of the quadratic reciprocity law in the language of quadratic groups. [251] P. Barrucand, F. Laubie, Sur les symboles des restes quadratiques des discriminants, Acta Arith. 48 (1987), 81–88 [252] T. Peklar, Varianta dokaza kvadratnega reciprocnostnega zakona (A proof of the quadratic reciprocity law) (Slowen.), Obt. Mat. Fiz. 36 (1989), 129–133 Tatjana PEKLAR proves the quadratic reciprocity law using GAUSS’s Lemma. [253] F. W. Barnes, A permutation reciprocity law, Ars. Comb. 29 A (1990), 155–159 BARNES studies the parity of the permutation on Z=nZ by mappings of the form x ! mx C c, and derives the quadratic reciprocity law in a similar way as ZOLOTAREV.

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[254] R.G. Swan, Another proof of the quadratic reciprocity law?, Amer. Math. Month. 97 (1990), 138–139 SWAN proves the quadratic reciprocity law by computing norms of certain elements in cyclotomic number fields. [255] G. Rousseau, Exterior algebras and the quadratic reciprocity law, L’enseignement Math. 36 (1990), 303–308 ROUSSEAU presents ZOLOTAREV’s proof in the language of exterior algebras of modules. [256] G. Rousseau, On the quadratic reciprocity law, J. Austral. Math. Soc. 51 (1991), 423–425 This proof became one of the best known proofs of the quadratic reciprocity law in recent years. [257] F. Keune, Quadratic reciprocity and finite fields, Nieuw Arch. Wisk. (4), 9 (1991), 263–266 KEUNE derives the quadratic reciprocity law from a computation of a Vandermonde determinant of a primitive q-th root of unity in a finite field of characteristic p. The proof is essentially the same as the one given earlier by D ELSARTE. [258] T. Kubota, The foundation of class field theory based on the principles of space diagrams (Japan.), Sugaku 44 (1992), 1–12; Engl. Transl. Sugaku Expo. 8 (1995), 1–12 KUBOTA sketches a geometric approach to explicit reciprocity laws in class field theory. The arguments he presents give a proof of the ordinary quadratic reciprocity law. [259] D. M. Russinoff, A mechanical proof of quadratic reciprocity, J. Autom. Reasoning 8 (1992), 3–21 RUSSINOFF describes a proof of the quadratic reciprocity law similar to GAUSS’s third proof obtained by the “Boyer-Moore theorem prover”. [260] P. Garrett, Quadratic reciprocity (following Weil?), online manuscript; URL: http://www. math.umn.edu/~garrett/index.shtml [261] K. Motose, On commutative group algebras, Sci. Rep. Hirosaki Univ. 40 (1993), 127–131 M OTOSE studies, for finite groups G and fields K, the algebra of mappings from G to K, and uses this tool for proving, among other results, the quadratic reciprocity law. [262] R. Laubenbacher, D. Pengelley, Eisenstein’s misunderstood geometric proof of the quadratic reciprocity theorem, College Mathematics Journal 25 (1994), 29–34 LAUBENBACHER & PENGELLEY present E ISENSTEIN’s geometric proof of the quadratic reciprocity law and compare it with G AUSS’s third proof. [263] G. Rousseau, On the Jacobi symbol, J. Number Theory 48 (1994), 109–111 ROUSSEAU uses ZOLOTAREV’s approach to the quadratic reciprocity law. [264] Ch. Cornaros, On Grzegorczyk induction, Ann. Pure Appl. Logic 74 (1995), 1–21 CORNAROS proves the quadratic reciprocity law in a fragment of Peano arithmetic. [265] P. T. Young, Quadratic reciprocity via Lucas sequences, Fib. Quarterly 33 (1995), 78–81 YOUNG proves the reciprocity law using formal group laws and LUCAS seqences; at bottom, it is a proof based on quadratic G AUSS sums. [266] J.-L. Brylinski, Central extensions and reciprocity laws, Cah. Topologie Geom. Differ. Categoriques 38 (1997), 193–215 [267] J.Y. Merindol, Symbole de Legendre et résultant, Revue de Math. Spéciales 107 (1996/97), 721–730 [268] T. Watanabe, Random walks on SL.2; F2 / and Jacobi symbols of quadratic residues, Advances in combinatorial methods and applications to probability and statistics (N. Balakrishnan, ed.), Birkhäuser, Statistics for Industry and Technology. 125–134 (1997) WATANABE presents a proof of the quadratic reciprocity law using ZOLOTAREV ’s approach. [269] M. Beck, The reciprocity law for Dedekind sums via the constant Ehrhart coefficient, Amer. Math. Monthly 106 (1999), 459–462 BECK observes that the reciprocity law for Dedekind sums implies the usual quadratic reciprocity law, and then proves DEDEKIND’s reciprocity law. [270] H. Ishii, Functional equations and the law of quadratic reciprocity, Mem. Inst. Sci. Engrg. Ritsumeikan Univ. No. 57 (1998), 1–3 (1999)

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15 Proofs of the Quadratic Reciprocity Law ISHII uses the fact that Gauss sums show up in the functional equations of Dirichlet L-series to give a proof of the quadratic reciprocity law based on functional equations of L-series. K. Motose, On commutative group algebras. III, Bull. Fac. Sci. Technol. Hirosaki Univ. 1 (1999), no. 2, 93–97 M OTOSE investigates the prime decomposition of quadratic Gauss sums and proves the quadratic, cubic and biquadratic reciprocity laws. M. Zahidi, Symboles des restes quadratiques et discriminants, thesis Limoges 1999 F. Lemmermeyer, Kreise und Quadrate modulo p, Math. Sem. Ber. 47 (2000), 51–73 This proof is a variant of LEBESGUE’s proof based on counting the number of solutions of quadratic congruences. J.L. Meyer, A reciprocity congruence for an analogue of the Dedekind sum and quadratic reciprocity, J. Théor. Nombres Bordeaux 12 (2000), 93–101 MEYER deduces quadratic reciprocity from a reciprocity law for sums similar to Dedekind sums. B.A. Tangedal, Eisenstein’s Lemma and quadratic reciprocity for Jacobi symbols, Math. Mag. 73 (2000), 130–134 TANGEDAL proves the quadratic reciprocity law using a Lemma due to E ISENSTEIN similar to GAUSS’s lemma. R. Chapman, email, 2000 In an email, C HAPMAN sent me a proof of the quadratic reciprocity law generalizing NAKASH ’s proof of .5=p/ D .p=5/ via Fibonacci numbers. K. Girstmair, Triple product identitites for the Jacobi symbol, Expo. Math. 19 (2001), 179–185 GIRSTMAIR first uses the quadratic reciprocity law for proving an identity showing up in the transformation of theta series; in a third part, he shows how to deduce this identity, from which the reciprocity law follows, in an independent way. L. Hammick, A simple proof of the quadratic reciprocity law, http://www3.telus.net/ldh/ math/qrl.html M.R. Murty, Quadratic reciprocity via linear algebra, Bona Mathematica 12, No. 4 (2001), 75–80 MURTY presents a proof of the quadratic reciprocity law using SCHUR’s determination of the sign of the quadratic Gauss sum. A.-M. Décaillot, Géométrie des tussus, Mosaïques. Échiquiers. Mathématiques curieuses et utiles, Revue Hist. Math. 8 (2002), 145–206 D ÉCAILLOT rewrites the proof published by LUCAS [94] in such a way that the connection with tissues mentioned by LUCAS becomes clear. M. Luo, A new elementary proof of the quadratic reciprocity law (Chin.), Sichuan Daxue Xuebao 40 (2003), 1–2 L UO presents a modification of Rousseau’s proof [256]. K. Motose, On Gauss sums and Vandermonde matrices, Bull. Fac. Sci. Technol. Hirosaki Univ. 6 (2003), no. 1, 19–23 K. Motose, see [282]. M OTOSE presents two proofs of the quadratic reciprocity law, both involving the discriminant of the cyclotomic polynomial and Vandermonde determinants. S.Y. Kim, An Elementary Proof of the Quadratic Reciprocity Law, Amer. Math. Monthly 111 (2004), no. 1, 48–50 KIM proves the quadratic reciprocity law along the lines of ROUSSEAU’s proof [256]. Zh.W. Sun, Quadratic reciprocity and power residues, preprint. see http://xxx.arxiv.cornell. edu/abs/math/0312010v1 partially published as Simple arguments on consecutive power residues, J. Number Theory 124 (2007), no. 1, 57–61 SUN rediscovers SCHOLZ’s proof of the quadratic reciprocity law. W. Duke, K. Hopkins, Quadratic reciprocity in a finite group, Amer. Math. Monthly 112 (2005)

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DUKE & H OPKINS show that a large part of the proofs of the quadratic reciprocity law based on G AUSS’s and ZOLOTAREV’s lemma can be formulated and proved as a result about quite general finite groups. M.R. Murty, A. Pacelli, Quadratic reciprocity via theta functions, Number theory, 107–116, Ramanujan Math. Soc. Lect. Notes Ser., 1, Ramanujan Math. Soc., Mysore, 2005 M URTY & PACELLI present Hecke’s proof of the quadratic reciprocity law in general number fields specialised to the field of rational numbers. M. Szyjewski, Algebraic proof of Gauss quadratic reciprocity, Int. J. Pure Appl. Math. 22 (2005), no. 2, 233–238 L.P. Arkhipova, On the quadratic reciprocity law (Russian), Chebyshevskii Sb. 7 (2006), 155–162 ARKHIPOVA presents a proof of the quadratic reciprocity law similar to GAUSS’s fourth proof. N. Robbins, Some properties of the inversion counting function, Georgian Math. J. 13 (2006), 783–786 ROBBINS denotes the sign in ZOLOTAREV’s Lemma by I.h; k/, so that .1/I.h;k/ D . hk / for coprime odd integers. Then he proves the relation h  I.h; k/ C k  I.k; h/ D 14 .h  1/.k  1/.h C k  1/, which implies the quadratic reciprocity law. S. Kumar, Theorema Aureum, Resonance 12 (2007), 25–39 K UMAR presents ROUSSEAU’s proof. S. Kumar, Theorema Aureum 2, Resonance 12 (2007), 18–29 S. Kumar, see [292]. KUMAR presents the proofs of the quadratic reciprocity law given by Keune [257] and by Swan [254]. W. Castryck, A shortened classical proof of the quadratic reciprocity law, Amer. Math. Monthly 115 (2008), 550–551 By a small modification, CASTRYCK succeeds in simplifying Lebesgue’s proof of the quadratic P i 2 reciprocity law based on counting solutions of congruences of the form a x  a mod p. S. Gurevich, R. Hadani, R. Howe, Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation, Int. Math. Res. Not. 2010, No. 19 (2010), 3729–3745 G UREVICH , H ADANI & H OWE give a proof of the quadratic reciprocity law using ingredients from W EIL’s proof in [212]. T. Kunisky, Quadratic reciprocity by group theory, Harvard College Math. Rev. 2, (2008), 75–76 In this article, high school student Tim K UNISKY presents his rediscovery of ROUSSEAU’s proof [256]. R. Jakimczuk, The quadratic equation in Fp and the quadratic reciprocity law, Int. J. Contemp. Math. Sci. 4 (2009), 419–431 JAKIMCZUK provides a variant of Lebesgues proof of the quadratic reciprocity law based on counting solutions of quadratic congruences. V. Schechtman, Sommes de Gauss et séries thêta, Ann. Fac. Sci. Toulouse 18 (2009), 663–683 SCHECHTMAN determines the sign of quadratic Gauss sums by analytic means and proves the quadratic reciprocity law following GAUSS’s fourth proof. S. Chebolu, J. Minác, C. Reis, Reciprocity laws for representations of finite groups, Ann. Sci. Math. Qué. 34 (2010), 37–61 CHEBOLU , MINÁC & REIS prove the quadratic reciprocity law using Vandermonde determinants and group representations. A. Kuroki, S. Katayama, A variation of Takagi’s proof for quadratic reciprocity laws of Jacobi symbols, J. Math. Univ. Tokushima 43 (2009), 9–23 KUROKI & K ATAYAMA present a modification of TAKAGI’s proof of the quadratic reciprocity law.

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[301] S. Hambleton, V. Scharaschkin, Quadratic reciprocity via resultants, Int. J. Number Theory 6 (2010), 1413–1417 HAMBLETON & SCHARASCHKIN prove the quadratic reciprocity law using resultants; the proof is a modification of SWAN’s proof [254]. [302] H. Verdure, A quadratic reciprocity law for elliptic curves, Acta Sci. Math. 75 (2009), 457–465 VERDURE proves an analogue of the quadratic reciprocity law for torsion points on elliptic curves, from which the usual quadratic reciprocity law follows. [303] E. Jerábek, Abelian groups and quadratic residues in weak arithmetic, Math. Log. Q. 56 (2010), 262–278 JERÁBEK proves the quadratic reciprocity law in “bounded arithmetic”. V ERDURE proves a “quadratic reciprocity law” for torsion points on elliptic curves and deduces the quadratic reciprocity law from it. This proof is related to the one based on comparing the splitting of primes in cyclotomic and quadratic number fields. [304] R. Steiner, Modular divisor functions and quadratic reciprocity, Amer. Math. Mon. 117 (2010), 448–451 STEINER proves the quadratic reciprocity law using calculations similar to those by ROUSSEAU [256]. [305] M. Szyjewski, Zolotarev’s proof of Gauss reciprocity and Jacobi symbols, Serdica Math. J. 37 (2011), 251–260 SZYJEWSKI presents a variant of Z OLOTAREV’s proof. [306] M. Dicker, A proof of the quadratic reciprocity law, available online: http://arxiv.org/abs/ 1210.7744 D ICKER proves the quadratic reciprocity law using the determinant of the matrix . ij /, where 0  i  p  1, 1  j  p, and where  is a primitive p-th root of unity. [307] S. Hambleton, V. Scharaschkin, Pell Conics and Quadratic reciprocity, Rocky Mountain J. Math. 42 (2012), 91–96 H AMBLETON & SCHARASCHKIN use division polynomials from the arithmetic of conics to prove the quadratic reciprocity law. [308] A. Karlsson, Applications of heat kernels on abelian groups: .2n/, quadratic reciprocity, Bessel integrals, Number theory, analysis and geometry, 307–320, Springer, New York, 2012 KARLSSON presents a version of G AUSS’s fourth proof of the quadratic reciprocity law. [309] R. Renko Zver, Dedekindove vsote in kvadratni reciprocitetni zakon (Dedekind sums and the quadratic reciprocity law), Obzornik mat. fiz. 59 (2012), 205–216 RENKO ZVER derives the quadratic reciprocity law from a reciprocity law for Dedekind sums. [310] M. Baker, J. Shurman, http://mattbakerblog.wordpress.com/tag/zolotarevs-lemma/ In some lecture notes, BAKER gave an interpretation of Z OLOTAREV’s proof of the quadratic reciprocity law; the version posted on his blog includes additional ideas by Jerry SHURMAN . [311] O. Demchenko, A. Gurevich, Reciprocity laws through formal groups, Proc. Amer. Math. Soc. 141 (2013), 1591–1596 D EMCHENKO & GUREVICH generalize HONDA ’s results and deduce various reciprocity laws. [312] Ph. Caldero, J. Germoni, Histoire hédonistes de groupes et de géométries, vol. I, Paris 2013 CALDERO and prove the quadratic reciprocity law by studying solutions of the PGERMONI congruences xi2  1 mod p.

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[313] Y. Burda, L. Kadets, Construction of the heptagecagon and quadratic reciprocity, C. R. Math. Rep. Acad. Sci. Canada 35 (1) 2013,16–21 BURDA and KADETS study the construction of the regular 17-gon using ruler and compass and prove the quadratic reciprocity law using the quadratic period equation over finite fields. [314] A. Brunyate, P. Clark, Quadratic reciprocity in abstract number rings, Ramanujan Journal (2014) BRUNYATE & CLARK investigate a group theoretical version of ZOLOTAREV’s Lemma.

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Author Index

Agou, 154 Aigner, 87, 152 Alexejevsky, 148 Allander, 155 Ankeny, 152 Arkhipova, 159 Arwin, 150 Auslander, 155

Bachet, 3 Bachmann, 87, 121 Baker, M., 160 Barbilian, 152 Barcanescu, 156 Barrucand, 144, 156 Bartelds, 150 Baumgart, 77, 143, 146, 147 Beck, 157 Berndt, 123, 155 Bhargava, 126 Birch, 87, 154 Bock, 146 Borchardt, 144 Bork, 145 Bouniakowski, 28, 29, 92, 95–97, 103, 143 Bouniakowsky, 130 Brandt, 152 Brenner, 154 Brewer, 152 Brown, 87, 156 Brunyate, 161 Burda, 161 Burde, 154, 155 Busche, 35, 37, 97, 98, 104, 130, 145–147, 149

Caldero, 160 Carlitz, 87, 153 Castaldo, 155 Castryck, 123, 140 Cauchy, 55, 115, 117–120, 122, 123, 129, 130, 140, 141, 144, 152 Chapman, 158 Chebolu, 159 Chowla, 154 Clark, 161 Cooper, v Cornaros, 157 Cuculière, 156

Décaillot, 158 Dedekind, 143, 144, 149 Delsarte, 152, 157 Demchenko, 160 Dicker, 160 Dickson, v Digby, 3 Dirichlet, 6, 12, 68, 85, 113, 115, 117, 119, 122, 125, 139–142, 144, 146 Dockeray, 151 Dörge, 151 Duke, 159

Eichenberg, 146 Eichler, 156 Eisenstein, 19, 41, 42, 55, 79, 80, 93, 94, 103, 107, 109, 119, 120, 129, 140, 141, 147–151, 154–156, 158 Ely, 140, 156

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6

167

168

Author Index

Euler, 3, 71, 77, 114, 139, 154 Evans, 123, 155

Ireland, 123 Ishii, 158

Fenster, v Fermat, 3, 139 Fields, 147 Fischer, 148 Frame, 155 Franklin, 146 Frenicle, 3 Friedlander, 155 Frobenius, 150, 151, 153 Fröhlich, 152 Fueter, 151 Furquim de Almeida, 152

Jacobi, xi, 6, 55, 74, 117–120, 127, 129, 140, 141 Jakimczuk, 159 Jenkins, 142 Jerabek, 160

Gauss, v, 4, 6, 7, 9, 12, 16, 18, 20, 27, 45, 48, 55, 61, 64, 72, 79, 81, 85–87, 90, 91, 93, 94, 100, 103, 111, 113, 117, 122, 125, 127, 139, 141, 144, 148, 152, 155 Gegenbauer, 81, 143, 145, 147 Genocchi, 20, 75, 94, 98, 102, 103, 123, 129, 130, 142, 144–146 Gérardin, 156 Germoni, 160 Gerstenhaber, 109, 154 Girstmair, 158 Goldschmidt, 156 Gurevich, 159, 160

Hacks, 146 Hadani, 159 Hambleton, 160 Hasenöhrl, 147 Hausner, 153 Heawood, 150 Heine, 115 Hensel, 144, 149 Hermes, 146 Hermite, 144 Hilbert, 148 Hirzebruch, 155 Holzer, 153, 154 Honda, 154, 160 Hopkins, 159 Howe, 159 Huber, 150 Hurrelbrink, 155 Husemöller, 155

Kac, 156 Kadets, 161 Kapferer, 151 Kaplan, 109, 154 Karlsson, 160 Katayama, 159 Keune, 157, 159 Kim, 104, 158 König, 87, 148 Koschmieder, 109, 154 Kronecker, 26, 75, 82, 86, 94, 98, 99, 103, 109, 112, 118, 119, 122, 123, 127, 143–148, 155 Kubota, 153, 154, 157 Kumar, 159 Kummer, 65, 67, 68, 125, 130, 139, 141, 151 Kunisky, 159 Kuroki, 159

Lagrange, 3, 118 Lange, 147, 148 Lasker, 144, 150 Laubenbacher, 157 Laubie, 144, 156 Lebesgue, 45, 61, 79–81, 115, 117, 118, 120, 122, 129, 140–142, 152, 154, 158 Legendre, v, 4–7, 87, 91, 127, 139, 148, 155 Lehmer, 87, 152 Lemmermeyer, 129, 140, 158 Lerch, 123, 146, 148, 149 Lewy, 151 Libri, 115 Liouville, 122, 129, 141, 144 Lipke, 87 Lucas, 146, 158 Luo, 158

Mansion, 143, 148 Mathews, 87 Mathieu, 142, 144 McClintock, 148

Author Index McDonnell, 150 Mertens, 147, 149 Meyer, C., 153 Meyer, J.L., 158 Milnor, 155 Minác, 159 Mirimanoff, 144, 149 Motose, 157, 158 Murty, 123, 158, 159 Nakash, 158 Olbers, 112 Pepin, 87 Pacelli, 159 Peklar, 156 Pellet, 144, 152, 156 Pengelley, 157 Pépin, 146, 148 Petersen, 33, 72, 94, 95, 103, 130, 144, 147 Petr, 149, 151, 152 Pieper, 127, 129 Plana, 141 Pocklington, 149 Poinsot, 118 Rademacher, 154 Rédei, 151–153 Reichardt, 153 Reis, 159 Renko Zver, 160 Reshetukha, 154 Riesz, 152 Robbins, 159 Rogers, 87, 155 Rosen, 123, 155, 156 Rousseau, 104, 157–160 Russinoff, 157 Ryan, 156 Salvadori, 123 Schaar, 123, 141, 142 Scharaschkin, 160 Schechtman, 159 Scheibner, 148 Scheja, 140 Schering, 21, 31, 75, 86, 90, 94, 98, 100, 103, 130, 143–146, 148 Schmidt, 147 Schönemann, 140 Scholz, 104, 151, 158

169 Schuh, 150 Schur, 123, 158 Serre, 109 Shanks, 123 Shapiro, 87 Shurman, 160 Skolem, 151–153 Smith, H.J., xi Sochocki, 144 Springer, 155 von Staudt, 143 Steiner, 160 Stern, 22, 23, 94, 103, 130, 143, 146, 147 Stickelberger, 144, 153, 156 Stieltjes, 150 Storch, 140 Sun, 158 Swan, 144, 153, 157, 159, 160 Sylvester, 78, 142 Szyjewski, 160 Tafelmacher, 146 Takagi, 149, 159 Tangedal, 158 Tate, 87, 154, 156 Teege, 87, 150 Thomae, 144 Tolimieri, 155 de la Vallée Poussin, 147 Vandiver, 153 van Veen, 151 Verdure, 160 Voigt, 34, 90, 91, 97, 103, 130, 144 Voronoi, 144, 150, 153, 156 Watanabe, 157 Waterhouse, 123 Weaver, 153 Weil, 154–156, 159 Whiteman, 151 Williams, K.S., 123 Young, 157 Zagier, 155 Zahidi, 144 Zantema, 156 Zassenhaus, 152 Zeller, 24, 94, 103, 130, 143–149, 153 Zolotarev, 143, 150, 153, 156, 157, 160

Subject Index

class ambiguous, 68 principal, 63 class field theory, 139 congruence binomial, 3

genus principal, 64 theory, 146, 148

Dedekind sum, 144, 153–158, 160 diophantine equation linear, 3

Jacobi sum, 56 multiple, 141

half-system, 15, 26, 72, 90

Legendre symbol, 102 Euclidean algorithm, 98

Fekete polynomials, 147 Fermat’s Theorem, 142 Fibonacci numbers, 158 form ambiguous, 63 class, 63 first kind, 63 positive, 63 primitive, 63 principal, 63 second kind, 63 formal group law, 157

Gauss sum, 48, 123, 140, 142, 144, 146, 152–155, 157–159 sign, 112, 140–142, 147, 148 Gauss’s Lemma, 19, 24, 140 in number fields, 153

Pell equation, 65, 125 period equation quadratic, 142, 144, 153

quadratic forms binary, 139 quadratic nonresidue, 3 quadratic period, 45 quadratic residue, 3

reciprocity law quadratic, 6, 7 Scholz, 140 resultant, 148, 160

supplementary law, 71 first, 73 second, 73, 147

© Springer International Publishing Switzerland 2015 O. Baumgart, The Quadratic Reciprocity Law, DOI 10.1007/978-3-319-16283-6

171

172

Subject Index

theorema fundamentale, 6 theta series, 112

Wilson’s theorem, 142

Vandermonde determinant, 152, 154, 157–159

Zolotarev’s Lemma, 159