Algebra: Volume I

B.L. van der Waerden

Algebra Volume I

Based in part on lectures by E. Artin and E. Noether Translated by Fred Blum and John R. Schulenberger

Springer

B.L. van der Waerden University of ZOrich (retired)

Pres.ent address: Wiesliacher 5 (8053) Zurich, Switzerland

Originally published in 1970 by Frederick Ungar Publishing Co., Inc., New York Volume I is translated from the German Algebra 1, seventh edition, Springer-Verlag Berlin, 196q. The work was first published with the title Modeme Algebra in 1930-1931.

Mathematics Subject Classification (2000): 00A05 01A75 12-01 13-01 16-01 20-01 ISBN 0-387-40624-7 First softcover printing, 2003. © 1991 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights ..

This reprint has been authorized by Springer-Verlag (BerlinIHeidelbergINew York) for sale in the People's Republic of China only and not for export therefrom. 987654321

SPIN 10947647

www.springer-ny.com Springer.. Verlag New York Berlin Heidelberg A member of Berte/smannSpringer Science + Business Media GmbH

FOREWORD TO THE SEVENTH EDITION

When the first edition was written, it was intended as an introduction to the newer abstract alge!>ra. Parts of classical algebra, in particular the theory of determinants, were assumed to be known. Today, however, the book is commonly used by students as a first introduction to algebra. It has therefore been necessary to include a chapter on "vector spaces and tensor spaces" in which the fundamental ideas of linear algebra, the theory of determinants in particular, are discussed. The first chapter, "Numbers and Sets," has been made shorter by treating ordering and well ordering in a new chapter (Chapt~r 9). Zorn's lemma is derived directly from the axiom of choice. A proof of the well ordering theorem is obtained with the same method (following H. Kneser). In the Galois theory certain ideas from the well-known book of Artin were adopted. A gap in a proof in the theory of cyclic fields, which several readers brought to my attention, was closed in Section 8.5. The existence of a normal basis is proved in Section 8.11. The first volume now concludes with the chapter "Real Fields." Valuation theory is presented in the second volume.

Zurich, February 1966

B. L.

VAN DER WAERDEN

FOREWORD TO THE FOURTH EDITION The algebraist and number theorist Brandt, who recently died unexpectedly, concluded his review of the third edition of this work in the Jahresbericht der D. M. V. 55 as follows: "As far as the title is concerned, I w()uld welcome it if the simpler, but more powerful title 'Algebra' were chosen for the fourth edition. A book which offers so much of the best mathematics, as it has been and as it will be, should not through its title give rise to the suspicion that it is simply following a fashionable trend which yesterday was unknown and tomorrow will probably be forgotten." Following this suggestion, I have changed the title to "Algebra." I am grateful for a suggestion by M. Deuring for a more appropriate definition of the concept of a "hypercomplex system" as well as an extension of the Galois theol7Y of cyclotomic fields~which seemed required with consideration of its application to the theory of cyclic fields. Many small corrections have been made on the basis of letters from various countries. I should here like to thank all writers for their letters. Zurich, March 1955

B. L.

VAN DER

WAERDEN

FROM THE PREFACE TO THE THIRD EDITION In the second edition the treatment of valuation theory was extended. It has meanwhile become more and more important in number theory and algebraic geometry. I have therefore made the chapter on valuation theory clearer and more detailed. In response to many wishes, I have again included the section on well ordering and transfinite induction, which was dropped in the second edition, and on this basis again presented the Steinitz field theory in complete generality. Following a suggestion of Zariski, the introduction of polynomials has been made easily comprehensible. The theory of norms and traces needed improve.. ment; Peremans had kindly pointed this out to me. Laren (Nordholland), July 1950

B. L.

VAN DER

W AERDEN

GUIDE The chapters of both volumes and their interrelation.

1 Sets

I

•9

I

2 Groups

Infinite Sets

3 Rinp

•

1

\

4

7

Groups

Vectors

•

1

I

S Polynomials

6 Fields

IS Ideal Theory ....._

...1....._

...

I

17 Integral Algebraic Quantities

-.......

1

I

I

I

8 Galois

12 Linear Algebra

Theory

•......- -.....- - - - - . - -..., I 10 Infinite Fields

18 Fields with Valuations 1-

I

16 Polynomial Ideals

I

I

11 Real Fields

19 Algebraic Functions

•

13 Algebras

I

14 Representation Theory

20

Topological Algebra

INTRODUCTION

PURPOSE OF THE BOOK The "abstract," "formal," or "axiomatic" direction, to which the fresh impetus in algebra is due, has led to a number of new formulations of ideas, insight into new interrelations, and far-reaching results, especially in group theory, field theory, valuation theory, ideal theory, and the theory of hypercomplex numbers. The principal objective of this book is to introduce the reader into this entire world of concepts. While, for this reason, general concepts and methods stand in the foreground, particular results which properly belong to classical algebra mu~t also be given appropriate co.nsideration within the framework of the modern development.

DISTRIBUTION OF SUBJECT MATTER. DIRECTIONS FOR mE READER In order to develop with sufficient clarity the general viewpoints which dominate the "abstract" approach to algebra, it was necessary to present afresh the fundamentals of group theory and of elementary algebra. In view of the fact that com~tent expositions on group theory, classical algebra, and the theory of fields have been published recently,l it was possible to present these introductory chapters briefly (but completely). Another guiding principle was the desire to make each individual part comprehensible in itself, insofar as this was possible. Those who wish to become acquainted with the general theory of ideals or with the theory of hypercomplex numbers need not study Galois theory beforehand and vice versa; those who want to consult the book about elimination or linear algebra need not be deterred by complicated ideal-theoretical terms. For this reason the subject matter has been distributed in such fashion that the first three chapters contain a most concise exposition of what is prerequisite to'all subsequent chapters: The fundamentals of the theories of 1. sets, 2. groups, IFor group theory the reader is referred to: Speiser, A.: Die Theorie der Gruppen von endlicher Ordnung, 2nd edition, Berlin. Julius Springer 1927 For the theory of fields: Hasse, H.: Hohere Algebra I, II and Aufgabensammlung zur Hoheren Algebra. Sammlung Goschen 1926-27. For classical algebra: Perron, 0.: Algebra I, II. 1927. For linear algebra: Dickson, L. E.: Modern Algebraic Theories, Chicago 1926.

X

INTRODUcnON

3. rings, ideals, and fields. The remaining chapters of the first volume are in the main devoted to the theory of commutative fields and are based primarily on Steinitz' fundamental treatise in erellea Journal, Vol. 137 (1910). The theory of modules, rings, and ideals with applications to algebraic functions, elementary divisors, hypercomplex numbers, and group representations will be treated in the second volume in several, mostly independent chapters. The theory of abelian integrals and the theory of continuous groups had to be omitted, since an appropriate treatment of both involves transcendental concepts and methods. Because of its extent, the theory of invariants could not be included, either. For further information we refer the reader to the table of contents, and especially to the foregoing schematic diagram which illustrates exactly how many of the preceding chapters are requisite to each of the chapters. The interspersed exercises may serve as a test whether the subject has become clear to the reader. Some of them contain examples and supplements, which are sometimes referred to in later chapters. No special devices are necessary for their solutions unless indicated in square brackets. SOURCES

This book has in part grown out of the following courses: Lectures given by E. Artin on Algebra (Hamburg, Summer session 1926). A seminar on Theory of Ideals, conducted by E. Artin, W. Blaschke, O. Schreier, and the author (Hamburg, Winter 1926-27). Lectures by E. Noether on Group Theory and Hypercomplex Numbers (Gattingen, Winter 1924-25 and Winter 1927-28)1. New proofs or new arrangements of proofs in this book are in most cases due to the lectures and seminars mentioned, regardless of whether the source is expressly quoted.

IAn elaborate treatment of the latter course by E. NoEtHER was published in MatA. Zeitschri/t Vol. 30 (1929), pp. 641-692.

CONTENTS

• Chapter 1

NUMBERS AND SETS 1.1 1.2 1.3 1.4 1.5

1

Sets 1 Mappings. Cardinality 2 The Number Sequence 3 Finite and Countable (Denumerable) Sets Partitions 10

7

Chapter 2

GROUPS

12

2.1 The Concept of a Group 12 2.2 SubgrQups 19 2.3 Complexes. Cosets 23 2.4 Isomorphisms and Automorphisms 25 2.S Homomorphisms, Normal Subgroups, and Factor Groups

Chapter 3

RINGS AND FIELDS

32

3.1 Rings 32 3.2 Homomorphism and Isomorphism 3.3 The Concept of a Field of Quotients 3.4 3.5 3.6 3.7 3.8

39 40

Polynomial Rings 43 Ideals. Residue Class Rings 47 Divisibility. Prime Ideals 51 Euclidean Rings and Principal Ideal Rings Factorization 57

53

28

XII

CONTENTS

Chapter 4

VECTOR SPACES AND TENSOR SPACES 4.1

4.2 4.3 4.4 4.5 4.6

4.7 4.8

61

Vector Spaces 61 Dimensional Invariance 64 The Dual Vector Space 66 Linear Equations in a Skew Field 68 Linear Transformations 69 Tensors 74 Antisymmetric Multilinear Forms and Determinants Tensor Products, Contraction, and Trace 80

76

Chapter 5

POLYNOMIALS 5.1 5.2

5.3 5.4

5.5 5.6

5.7 5.8

5.9 5.10

83

Differentiation 83 The Zeros of a Polynomial 85 Interpolation Formulae 86 Factorization 91 Irreducibility Criteria 94 Factorization in a Finite Number of Steps 97 Symmetric Functions 99 The Resultant of Two Polynomials 102_ The Resultant as a Symmetric Function of the Roots Partial Fraction Decomposition 107

105

Chapter 6

THEORY OF FIELDS 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

110

Subfields. Prime Fields 110 Adjunction 112 Simple Field Extensions 113 Finite Field Extensions 118 Algebraic Field Extensions 120 Roots. of Unity 125 129 Galois Fields (Finite Commutative Fields) Separable and Inseparable Extensions 133 Perfect and Imperfect Fields 137 Simpl!city of Algebraic Extensions. Theorem on the Primitive Element , 139, 6.11 Norms and Traces 140

Contents

xm

Chapter 7 CONTINUATION OF GROUP THEORY 7.1

7.2 7.?t 7.4 7~5

7.6 7.7 7.8 7.9

144

Groups with Operators 144 Operator Isomorphisms and Operator Homomorphisms The Two Laws of Isomorphism 147 Normal Series and Composition Series 148 Groups of Order p" 152" Direct Products 153 Group Characters 156 Simplicity of the Alternating Group 160 Transitivity and Primitivity 162

146

Chapter 8 THE GALOIS THEORY

8.1 8.2 8.3 8.4 8.5 8.6 8.-7 8.8 8.9 8.10 8.11

165

The Galois Group 165 The Fundamental Theorem of the Galois Theory 168 Conjugate Groups, Conjugate Fields, and Elements 170 Cyclotomic Fields 172 Cyclic Fields and Pure Equations 178 Solution of Equations by Radicals 181 The General 'Equation of Degree n 184 Equations of the Second, Third, and Fourth Degrees 187 Constructions with Ruler and Compass 193 Calculation of the Galois Group. Equations with a Symmetric Group 197 Normal Bases 200

Chapter 9 ORDERING AND WJ:LL ORDERING OF SETS 9.1 9.2 9.3 9.4

Ordered Sets 205 The Axiom of Choice and Zorn's Lemma The Well-Ordering Theorem 209 Transfinite Induction 209

206

205

XIV

CONTENTS

Chapter 10 INFINITE FIELD EXTENSIONS 10.1 10.2 10.3 10.4 10.5

212

Algebraically Closed Fields 212 Simple Transcendental Extensions 217 Algebraic Dependence and Independence 220 The Degree of Transcendency 223 Differentiation of Algebraic Functions 225

Chapter 11 REAL FIELDS 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Index

231

Ordered Fields 231 Definition of the Real Numbers 234 Zeros of Real Functions 242 246 The Field of Complex Numbers Algebraic Theory of Real Fields 248 Existence Theorems for Formally Real Fields Sums of Squares 256 258

253

Chapter 1

NUMBERS AND SETS

We begin with a brief chapter on certain logical and general mathematical concepts, especially that of a set, which are used throughout this book and are often unfamiliar to a beginner. We shall not go into the complications of the foundations of mathematics; we shall adopt a "naive viewpoint" throughout, yet avoid circular definitions which lead to paradoxes. The more advanced student need remember only the meaning of the symbols E, ~, ::>, n, V, and {...}, as explained in this chapter and may skip the rest of it.

1.1 SETS As a starting point for all mathematical considerations we take certain objects, such as numerals, letters, or their combinations. A set or a class is defined by any property which each one of these objects does or does not have. Those objects which have this property are called elements of the set. The symbol . a E IDl

means: a is an element of IDl; geometrically speaking, we say: a lies in IDl. A set will be called empty if it does not contain any elements. We assume that it is legitimate to regard sequences and sets of numbers again as objects and elements of sets (sets of second order, as they are sometimes called). These sets of second order may again be elements of sets of a higher order. However, we shall avoid such terms as "the sets of all sets," since they may give rise to contradictions (and have done so in the past); we shall rather form new sets only from a previously strictly defined category of objects (to which the new sets do not belong). If all elements of a set mare at the same time elements of IDl, then 91 is called a subset (subclass) of un, and we write:

91

~

un.

Also ID1 is said to include or contain 9l, and we write:

IDl If

~ ~ ~

and

~ ~ (£,

then

~

C

;2

9l.

(t.

1

2

NUMBERS AND SETS

The empty set is contained in every set. If all elements of IDl are in 9l, and all elements of ~ are in ill, then the sets in and 91 are said to coincide or to be equal:

rot

= 91.

Thus the equality IDl = 91 means that the relations

ill

~

91,

9l c IDl

hold simultaneously. Or we may say: Two sets are equal if they contain the same elements. If ~ IDl without being equal to IDl, is called a proper subset of IDl, or we say: contains 91 properly. We write

m

m

m

~

c: IDl,

IDl

::>

91.

Thus, 9l c IDl means that all elements of 91 are in in and that there is at least one more element in ill not belonging to 9l. Let ~ and $ be two arbitrary sets. The set 1) consisting of aU elements common to ~ and ~ is known as the intersection of ~ and ~, and we write 1)

t>

= [~,

~]

= ~ n ~.

is a subset of both ~ and ~. Any set having this property is contained

in D. The set ~ consisting of all elements that .belong to at least one of the sets 9I and ~ is called the union of ~ -and ~: $

=~

v~.

~

includes both ~ and ~, and any set including ~ and ~ includes ~ as well. The same definitions. hold for the intersection and union of an arbitrary set ~ of sets ~, ~, .... For the intersection (that is, the set of the elements wh.ich lie in all sets ~, ~, . . . of the set ~) we write:

= [91,

B, · · . ] . Tw~ sets are said to be mutually exclusive ~r disjOint if their intersection is empty, that is, if the two sets have no common elements. If a set is given by the enumeration of its elements (for example, if we say the set IDl is to consist of the elements a, b, c), we write: 1) (l:)

IDl = {a, b, c}. This notation is justified, since, according to the definition of equality of sets, a set is determined by its elements~ The defining property characterizing the elements of in: is: to be identical with a or b or c.

1.2 MAPPINGS. CARDINALITY If, by some rule, to each element of a of a set Wl there corresponds a single new object q>(a), then the correspondence tp is called a function. If all the new

The Number Sequence

3

objects {a) is called a mapping from IDl to 91. The element b the number 0 if a

=b

the negative integer - (b-a) if a 1 generate the symmetric group 6 n • 2.10. Prove, as in 2.9, that for n> 2 the n - 2 cyclic permutations on three digits (1 2 3), (1 2 4), ... , (1 2 n) generate the alternating group m:n• We shall now determine all subgroups of the cyclic groups. Let OJ be a cyclic group generated by a, and let 9 be a subgroup not consisting of the identity alone. If 9 contains an element a- m with a negative exponent, it will also contain the inverse element alii. Let am be the element of 9 having the smallest positive exponent. We shall prove that all elements of 9 are powers of am. If a' is an arbitrary element of g, we have

s = qm+r

(O~r<m).

Now t/(am)-q = d- mq = d is an element of g, where r<m, whence it follows that r = 0 because of the choice of m, and therefore s = qm and t/ = (amy'. Thus all elements of 9 are powers of am. If a is of finite order n so that ti' = e, then n must be divisible by m: n = qm, since d' = e is in the subgroup g. The subgroup 9 then consists of the elements a"', a2111 , ••• , film = e and is of order q. If, on the other hand, a is of infinite order, the subgroup 9 consisting of the elements e, a±'", a±2"" ... is likewise of infinite order. Thus we have proved the following.

Co~kx~.Couu

n

A subgroup of a cyclic group i9 itself cyclic. It consists either o/just the 1, or of the powers of the element am with the smallest possible positive m; in other words, it consists of the mth powers of the elements of the original group. For a cyclic group 0/ infinite order, m mIly be chosen at will, whereas .for a cyclic group of finite order n the number m must be a factor of n. In this case the subgroup is of order q = n/m. To every such number m belongs one, and only one, subgroup {aIN} of the cyclic group {a}.

2.3 COMPLEXES. COSETS In group theory a complex is defined as an arbitrary set of elements of a group ffi. By the product gl) of two complexes 9 and I) we understand the set of all products gh where g is taken from g, and h from I). If in the product gf;) one of the complexes, 9 for example, consists of only one element g, we may simply write gl) instead of gf). Obviously, the- following rule holds: g~f)

= (gl)f.

In composite products of complexes the parentheses may therefore be omitted [cf. (2.1)]. If the complex g is a group, then gg

= g.

Let 9 and 1) be subgroups of ffi. The question arises: Under what conditions is the product 99 itself a group? The totality of the inverses of the elements of 99 is 1)g, since the inverse of gh is h -1 g -1. Thus, for 91) to be a group, it is necessary that

l)g

= gl);

(2.5)

that is, 9 must commute with 1). This condition is also sufficient, for if it is satisfied, gl) contains for every gh also the inverse h- 1g- 1 and, moreover, for any two elements also their product, since gl)gl) = 991)9 = gl). Restating the above, we say: The product 99 of two subgroups g and 1) of OJ is itself a group if and only if the subgroups 9 and ~ commute. It is of course not necessary that each element of g commute with each element of I). If condition (2.5) is satisfied, then the product gl) is the group generated by 9 and~. In an Abelian group, (2.5) is always satisfied. If we write the Abelian group under the law of addition, that is, if 9 and 1) are submodules of a module, we write (g, 1) instead of 91), and we shall reserve the symbol 9 +9 for the special case of the "direct sum" to be investigated later.

24

GROUPS

H 9 is a subgroup, and a an element of (f), then the complex ag is called a left coset, and the complex ga a right coset (or a residue class) of gin (fj. If a lies in g, we have ag = g; hence one of the left (and also one of the right) cosets of 9 is always equal to 9 itself. In the following we shall mainly be concerned with left cosets, although all our considerations are valid for right cosets as well. Two co sets ag, bg can very well be equal without a being equal to b; for whenever a- 1 b lies in g, we have bg

= aa-1bg = a(a- 1bg) = age

Two different cosets have no common element; for if the cosets ag and bg have a common element, say

it follows that

glg2 -1 = a- 1b,

so that a-1b lies in g; accordingly, ag and bg are identical. Every element a belongs to a coset, namely to the coset age The latter surely contains the element ae = a. According to what has just been proved, the element Il belongs to just one coset. Therefore we may regard any element a as a repre.. M!1Jtative of the coset ag containing Q. According to the preceding paragraphs, the eosets constitute a partition of the group O) may be represented by the numbers 0, I, ... ,m-I and may thus be denoted by .Ro, .Sl1' · • • , .R"._ 13.14. What ideal is generated by the numbers 10 and 13 together in the ring of integers? 3.15. What does a == b(O) mean? 3.16. All multiples ra of an element a form an ideal oa. Considering the ring of the even integers, make clear to yourself that this ideal is not necessarily identical with the principal ideal (a). Ideals bear the same relation to ring homomorphism as do norma] subgroups to group homomorphism. Let us start from the notion of homomorphism. A homomorphism 0 "......, 0 between two rings defines a partition of the ring 0: a class 5\" is formed by all elements a having the same image a. We can characterize these classes more precisely, as follows. The class n of 0 to which corresponds the zero element under the homomorphism o ".. . ., 0 is an ideal in 0, and the other classes are the residue classes of this ideal. Proof: In the first place n is a module, for if a and b become zero under the homomorphism, so does -b, and hence also the difference a-b; thus, if a and b belong to the class n, so does a-b. Also, n is an ideal, for if a is mapped into zero, and if r is arbitrary, ra is mapped into f-O = 0, thus belonging to n again. n is thus a two-sided ideal in o. The elements a + c(C En) of a residue class modulo n with a as its representative are carried into ti + 0, hence into d; therefore, all of them belong to a class 5\a. If, conversely, an element b is carried into a, then b-a becomes a-a = 0; hence b - a E n, and b lies in the same residue class as Q. This completes the proof. Thus, to every homomorphism belongs an ideal. And now for the converse. We start from an ideal m in 0 and ask whether there exists a ring 0 homomorphic with 0, so that the elements of ij correspond exactly to the residue classes modulo m. In order to construct such a ring, we proceed in the same ~anner as we did

so

RINGS AND FIELDS

in Section 2.5: As elements of the ring to be constructed we simply choose the residue classes modulo m, denote the residue classes a+m by a, and try to define for them an addition and multiplication so that the correspondence a~d constitutes a homomorphism. Thus, for any two residue classes a, h, we have to determine a sum class Ii + b and a product class Ii· b so that all the sums of the elements of d and those of h are in the sum class and all products in the product class. Thus let a be an element in ti, and b one in b. We tentatively define d + b as the class containing a+b, and a" b as the class containing a·b. If a' a is some other element of a, and b ' = b one of b, then, according to the above, 8

=

a'+b'

=a+b

a'·b'

= a'b;

hence a' +b' is in the same residue class as a+b, and a' ·b' in the same as a-b. Our definition of the sum class and product class is therefore independent of the choice of the elements a, b within a, h. Now these classes a+ h, a· b have the desired property, namely that every sum a' +b' is in the sum class d+ h, and that every product a'· b' is in the product class of a' h. To every element a there corresponds a residue class d; this correspondence is a homomorphism, since the sum a+b corresponds to the sum a+b, and the product dh likewise to abo Hence the residue classes form a ring (Section 3.2). We shall call this ring the residue class ring o/m of 0 modulo m. The ring 0 is homomorphically mapped upon o/m by means of the correspondence mentioned before. For this homomorphism the ideal m plays the same role as n above; it is indeed identical with the set of all elements whose residue class is the zero class. We note the fundamental importance of the ideals: they enable us to con,struct rings homomorphic with a given ring. The elements of such a new ring are the residue classes modulo an ideal: with every element a there is associated a residue class d. Two residue classes are added or multiplied by adding or multiplying any two representatives of these residue classes. Since a = b implies d = b, the transition to the residue class ring transforms congruences into equalities. The special rings homomorphic with 0, as here constructed, that is, the residue class rings o/m, essentially exhaust all rings homomorphic with o. For if 0 is an arbitrary homomorphic image of 0, there is, as we have seen, a biunique correspondence between the elements' 'of 0 and the residue classes of 0 modulo an ideal n in o. The element d in i.i corresponds to the residue class .Ra • The sum and the product of two residue classes .Ra , .Rb are given by .Ra + band .Rab , respectively, and to the latter correspond, respectively, the elements

a+b

= a+b

and ab = 8All

congruences are of course modulo m.

dh.

Divisibility. Prime/deals

51

Hence the correspondence of the residue classes to the elements of D constitutes an isomorphism. Thus we have proved the following. Any ring 0 homomorphic with 0 is isomorphic with a residue class ring o/n, where n is the ideal of those elements whose image in '0 is the zero. Conversely, every residue class ring o/n is a homomorphic image 0/0. (Homomorphism theorem lor rings). EXAMPLES OF A RESIDUE CLASS RING In the ring of integers we may denote the residue classes modulo a positive number m by Ro, R l' ••. , .R m - 1 (cf. Exercise 3.13), where .R" consists of those numbers which, upon division by m, leave a remainder a. In order to add or multiply two residue classes R 5l: b , we add or multiply their representatives a, b, and reduce the result to its least nonnegative remainder modulo m. Q ,

Exercises 3.17. The residue class ring olm may have divisors of zero, even though 0 does not have any. Give examples in the ring of integers. 3.18. The homomorphism 0 0 is an isomorphism if and only if n = (0). 3.19. In a field there are no ideals except for the null ideal and the unit ideal. Furnish the proof. What does this imply for the possible homomorphic mappings of a field? f'J

3.6 DIVISmILITY. PRIME IDEALS

:Let b be an ideal (or, more generally, a

module) in the ring o. If a is an element ofb, we may write a O(b), and we say that a is divisible by the ideal' b. If all the elements of an ideal (or of a module) Q are divisible by b, then a is called divisible by b; this simply means that a is a subset ofb. In symbols:

=

a = O(b). We call Q a multiple or, in modern terminology, a subideal ofb. Similarly, b is called a divisor or an overideal of Q. If moreover Q =f= b, then b is called a proper divisor of Q and Q is called a proper multiple of b. For principal ideals in commutative rings with identity, (a) = simply means a = rb, and the concept of divisibility as defined by ideals becomes identical with the ordinary concept. From now on all rings under consideration will again be commutative. An i13) < N(ft).

(t)2+(l)2. A basis t 1, ••• ,qr of 1, p prime) is always irrational.

96

POLYNOMIALS

EXlUllple 2: f(x) = x p - 1 +Xp - 2 + _ •• +1 is the left member of a "cyclotomic equation" if p is a prime number. We again ask for irreducibility over the ring of integers. The Eisenstein criterion cannot be applied directly, but we can reason as follows: Iff(x) were reducible, f(x+ 1) would be also. Now we have

j'(x+l)

,

= (x+l)"-l =

x,+(P)X'-l+ _. -+ ( P )x 1

p-l

x

(x+l)-l

= x,.-l + (~)xP-2+. ··+ (p~ 1)' All coefficients, except that of xl'-1 are divisible by p; for in the formula for the binomial coefficients

(p)i =p(p-l) ...i! (p-i+l)

the numerator is divisible by P for i 1. 5.13. Show that x2+y3_1 is irreducible in P[x, y], where P is any field in which 5.14. Show that the polynomials

+ 1 =1=

-1.

are irreducible in the polynomial domain over the integers. Basically, the Eisenstein theorem rests on the fact that the equation

f(x)

= g(x)-h{x)

is transformed into a congruence modulo p1., namely f(x)

= g(x)· h(x),

Factorization in a Finite Number of Steps

97

which leads to an absurdity. In many other cases it is likewise possible to furnish irreducibility proofs by transforming the equations into congruences, modulo some quantity q of the domain 6, and by investigating whether the polynomial f(x) under consideration can be resolved modulo q. If, in particular, ($ is the domain of the integers I, then there are only a finite number of polynomials of a given degree in the residue class domain modulo q; hence there are always but a finite number of possibilities of a resolution of f(x) modulo q that have to be investigated~'Ifitis found that/ex) is irreducible modulo q., thenf(x) was also irreducible in l[x], and even in the opposite case we might be able to draw conclusions from the decomposition modulo q. In the case where q is a prime number we may apply the unique factorization theorem of the polynomials modulo q (Exercise 3.27). Example 4: 6 = l ;f(x) = x 5 _x2 + 1. Iff(x) factors modulo 2, then one of the factors has to be linear or quadratic. Now there are but two linear polynomials modulo 2: x, x+ 1,

and but one irreducible quadratic polynomial:

x2+x+l. On performing the division, we see that X S _x2 +1 is not divisible by any of these polynomials (modulo 2). This can be seen directly from x 5 _Xl+ 1

= x 2(x3 -1)+ 1 == Xl(X+ 1) (Xl+X+ 1)+ 1.

Hence, f(x) is irreducible.

5.6 FACTORIZATION IN A FINITE NUMBER OF STEPS Thus far we have only seen that there is a theoretical possibility to decompose into prime factors any polynomial in 1; [Xl' ... , XII] for a given field l:, and in some instances ,we have provided the tools for actually furnishing a decomposition, or for showing the impossibility; yet we still lack a general method for performing the factorization in a finite number of steps for any case that may presen~ itself to us. We proceed to develop such a method at least for the case in which ~ is the field of rational numbers. According to Section 4.5, we may assume the coefficients of any rational polynomial to be integers, and we may perform its factorization in the domain of integers. In the ring l of the integers itself a factorization into primes can evidently be performed by a finite trial and error method; furthermore, there are only a finite number of units ( + 1 and -1) in the ring I, and hence a finite number of possible factorizations. Similarly, in the polynomial domain Z[Xl' •.• ,xJ there are only the units + 1, -1. By the method of induction on the variable number n we shall now reduce everything to the following problem.

98

POLYNOMIALS

Let any factorization in 6 be performable in ajinite number of steps: moreover, let there be only afinite number of units in 6. We wish tofind a method offactoring every polynomial in S[x] into prime factors. The solution is due to KroneCker. Letf(x) be a polynomial of degree n in 6[x]. Iff(x) can be factored, then one of the factors is of degree ~ n12; thus, if s is the greatest integer ~ n12, we must investigate whether f(x) has a factor g(x) of degree ~ s. We form the functional values {(ao), /(a1)' ... ,/(as) for s+ 1 integral argu.. ments ao, aI' ... , as. Iff(x) is to be divisible by g(x), thenf(ao) must be divisible byg(ao), andf(al} by g(at), and so on. However, every[(aj) in 6 possesses only a finite number of factors; therefore, for every g(aJ there are only a finite number of possibilities all of which may be found explicitly. For every possible combination ofvaluesg(ao), g(al), ... , g(as) there is, according to the theorems of Section 4.4, one and only one polynomial g(x) which may be formed by Lagrange's or, more conveniently, Newton's interpolation formula. In this way a finite number of possible factors g(x) are found. Employing the division algorithm, we may now find out whether each of these polynomials g(x) is actually a factor of f(x). H, apart from the units, none of the possible g(x) is a factor of f(x) , then f(x) is irreducible; otherwise, a factorization has been found, and we may proceed to apply the same procedure to the two factors, and so forth. In this manner we finally arrive at the irreducible factors. In the integral case (6 = l) the procedure may frequently be shortened considerably. By factoring the given polynomial modulo 2 and possibly modulo 3, we get an idea what degrees the possible factor polynomials g(x) might have, and to what residue classes the coefficients modulo 2 and 3 might belong. This limits the number of the possible g(x) considerably. Moreover, when applying Newton's interpolation formula, one should note that the last coefficient As must be a factor of the highest coefficient of f(x), which limits the number of possibilities still further. Finally, it is an advantage to use more than s + 1 points a j (preferably 0, ± 1, ±2 and so on). For determining the possible g(ai) we use those !(a i ) which contain the least number of prime factors; the other points may afterwards be used in order to limit the number of possibilities still further by examining each g(x), and to see whether it assumes values which are factors of the respective f(a i ) at all points ai. >

Exercises

5.15. Factor in l[x]. 5.16. Factor f(x, y, z)

in l[x, y, z].

= -x3 _y3_ z 3 + x 2 (y +z) +y2(X+Z)+ z2(x+y)-2xyz

Symmetric Functions

99

5.7 SYMMETRIC FUNCTIONS Let 0 be an arbitrary commutative ring with an identity element. A polynomial in O[Xl' ••• ,XII] which is unchanged by any permutation of the indeterminates Xl, ••• , X. is called a (rational integral) symmetric function of the variables Xl"'" x". Examples: sum, product, sum of powers sQ = Ll'!lXy", Introducing a new indeterminate i, we put

J(z)

= (Z-Xl) (Z- X2) .•. (z- XII) = z"-ulz"-l +u2z"-2-

• · • +( -1)"0'".

(5.17)

The coefficients of the powers of z in this polynomial are 001

= Xl +Xl+'"

+ XII'

= XtXl +XIX3 + · · · +X2 X 3 + · · · +X,,-t X ,., (73 = XIX2X3+X1X2X4+'" +X,.-2 X ,.-lX", 0'2

Obviously, they are all symmetric functions, since the left side of (5.17) remains unchanged by any permutations of the Xi' We call ai' ... , a,. the elementary symmetric functions of Xl' ••• , XII' A polynomial tp{O'H ••• , uJ becomes a symmetric function of the Xl' ••• , XII when the a are written in terms of the x. Thus a term CO't"l ••• a,,"" of cp(0'1, ••• , a,J becomes a homogeneous polynomial in the Xi of degree P,1 + 2P,2 + · · · + np,,,, since every 0' i is a homogeneous polynomial of the ith degree. The sum ILl + 2P.2 + · · · +np." will be called the weight of the term Calill ... a:". The weight of a polynomial 9J{ul ' •.• , a,.) is defined as the largest weight occurring among its terms. Polynomials 2 = f(J, we see that it is sufficient to show that rot, · · · ,Yft) =t= 0 implies tp{0'1' • • · , O'J =t= O. Proof: Each term of q>(Yl' ••• ,Yft) can be written in the form -.(Xl ,,«2 -113 a",«l )'1 )'2

• ••

y

tin

n

•

Among all systems (ai' !X2, ••• , a,,) belonging to coefficients a =t= 0, there is a first one in the lexicographic ordering. If we replace the Yi by the O'i and express these in terms of the X" then we obtain as the lexicographically first term of CP(0'1' • • · , un) ax 1(11

•••

X «,. "

•

This term cannot be canceled, and thus

q.(0'1' · •• , uJ =t=

o.

Symmetric Functions

101

We have thus proved the following. A symmetric polynomial in O(X 1 , ••• , xJ may be written in one, and only one, way as a polynomial in at, ... , a,.; the weight of this polynomial is equal to the degree of the given one. All integral rational relations between symmetric functions are preserved, if the Xl are not indeterminates, but are quantities iI). 0, for example, the roots of a polynomialf(z) completely decomposable in o[z]. Thus it follows from what has been proved that a symmetric function of the roots of J(z) may be expressed in terms of the coefficients of J(z).

Exercises 5.17. For arbitrary n, express the "sums of powers" LXb LXi2, LXi3 by the elementary symmetric functions. S.lS. Let LX,' = Sf. Prove the formulae

s,-s'-lal +9,-20'2 - · · · +( -l),-lSl a ,_l +( -l)'eO'e = 0 sQ-sQ-lal + ... +(-I)"sQ_"O''' = 0 Employing these formulae, express the sums of powers in terms of elementary symmetric functions.

for e~n for e>n.

SI' S2' 33' S4, 35

An important symmetric function is the square of the difference product: D

=

n (Xj-X,,)2.

f is the ideal of those polynonuals f(x) which have {) as a root, that is, for whichf({) = O. Since (5 has no divisors of zero, ~[x]/p cannot have any either; hence the ideal p must be prime. Furthermore, p cannot be the unit ideal, since under the homomorphism the identity e is not associated with the zero but with e itself. Since every ideal in d[x] is a principal ideal, there remain but two possibilities: 1. l> = (cp(x)), where cp(x) is a polynonlial irreducible in a[x]. 2 , (h)

= q>(r 1)CP(r2) ... cp(r".);

6According to Exercise 3.24, cp(h) is also the number of the natural numbers < h relatively prime to h. rp(h) is called Euler's cp-function.

128

THEORY OF FIELDS

hence, by (6.3),

q;(h)

= q~1-1(ql -1)qi2 -1(q2 -I) ... q:rm-1(qm -1)

=h(l-:)(1-:J . -(l-:J.

Thus we have:

The number of the primitive hth roots 0/ unity is CP(h)

= h fi1 (1-~). qi

We put n = q;(h). Let the primitive hth roots of unity be are the roots of the polynomial

'1' ".. ,'JI.

They

(x-'t) (X-'2) · •• (x-',,) =