Modular Invariants

Cambridge Tracts in Mathematics and Mathematical Physics GENERAL EDITORS

HARDY, M.A., CUNNINGHAM,

G. H. E.

F.R.S.

M.A.

No. 27

MODULAR INVARIANTS

London Cambridge University Press Fetter Lane

New

York, Toronto

Bombay, Calcutta, Madras Macmillau Tokyo

Maruzen Company Ltd

All rights reserved

MODULAR INVARIANTS

BY D. E.

RUTHERFORD,

B.Sc.

CAMBRIDGE AT THE UNIVERSITY PRESS 1932

PRINTED IN GREAT BRITAIN

PREFACE the winter of 1929 Professor Weitzenbock pointed out to

me

that

INthere was no complete account of the theory of modular invariants embodying the work of Dickson, Glenn and Hazlett. The sole source of information on this subject was a number of papers, most of which appeared in American periodicals, and a tract by Dickson which contained the substance of his Madison Colloquium Lectures. giving a good account of the subject as

it

This tract, while

was understood in 1914, was

published before the modular symbolical theory was instituted. Although the symbolical theory

is

not yet complete,

certainly affords a

it

much

better introduction to the subject than did the earlier non-symbolical

methods. The theory

is

much hampered by

which seem to be true but

the lack of two theorems

no proof has been given. These are (i) that all congruent covariants can be represented symbolically; (ii) that Miss Sanderson's theorem can be applied to for which, as yet,

covariants as well as to invariants.

In preparing the present account, the chief difficulty has been the

method

lack of any systematic

of approach, since

most of the papers on My aim

the subject have been concerned with particular cases only.

has been to give a clear and concise account of the theory rather than to give a complete survey of the subject, in this tract only those

For the sake of completeness tricate proof of Dickson's this

and

methods which seem it

I

has been necessary to include the in-

theorem in paragraph

might be omitted at a

first

have therefore included

to be of general application.

reading.

13.

It is

suggested that

In order to avoid confusion

the reader should notice that the words fundamental and modular vary

somewhat in meaning I

in the different papers on the subject.

have, of course, benefited considerably from the papers of Dickson,

Glenn, Hazlett, Sanderson and others, and

many theorems

are taken

directly from their papers. The substance of Part II is largely taken " from a course of lectures entitled Algebraische theorie der lichamen"

which Professor Weitzenbock delivered in Amsterdam University during

PREFACE

VI the session 1929-30.

I

he has kindly placed at of great assistance to helpful advice.

My

have also made use of his lecture notes which

my

me

disposal.

throughout

Professor Weitzenbock has been

my

grateful thanks are

work and has given me much due to Professor Turnbull of

St Andrews University and to Professor Weitzenbock for reading the proof-sheets

and

for

making many suggestions and

Many thanks are also due

to the Syndics of the

corrections.

Cambridge University

Press for their helpful criticism of the manuscript.

D. E. ST ANDKEWS April 1932

K

CONTENTS

Preface

Contents

.......... ........ PART

SECT. 1

.

.

r

A new notation

page v vii

I

.... .... ....... .... ....

1

2.

Galois fields and Fermat's theorem

1

3

3.

Transformations in the Galois

4.

Types of concomitants

5.

Systems and finiteness

6.

Symbolical notation

fields

.

7.

Generators of linear transformations

8.

Weight and isobarism

9.

Congruent concomitants

4 6 6

8

...... ....... ........22

10 10

10.

Relation between congruent and algebraic covariants

11.

Formal covariants

12.

Universal covariants

13.

Dickson's theorem

.

12 15 15

17

14.

Formal invariants of the linear form

15.

The use

16.

Annihilators of formal invariants

of symbolical operators .

.

.

.... ...

24 26

17.

Dickson's method for formal covariants

18.

Symbolical representation of pseudo-isobaric formal Co-

19.

Classes

31

20.

Characteristic invariants

33

21.

Syzygies

35

22.

Residual covariants

36

23.

Miss Sanderson's theorem

39

.........

variants

24.

A method

25.

Smallest

26

Residual invariants of linear forms

of finding characteristic invariants

full

systems

.

.

....

28

30

42 43 45

CONTENTS

VU1 BECT.

27.

Residual invariants of quadratic forms.

28.

Cubic and higher forms

.

.

.

page 47 51

29.

Relative unimportance of residual covariants

30.

Non-formal residual covariants

PAET

.

.

51

51

II

31.

Rings and

32.

Expansions

54

33.

Isomorphism

57

34.

Finite expansions

35.

Transcendental and algebraic expansions

36.

Rational basis theorem of E. Noether

52

fields

Kpf

37.

The

38.

Expansions of the

39.

The theorem on

40.

jR-modules

fields

first

and second

...

....

sorts

...

65 66

70

41.

A theorem The

43.

Application of E. Noether's theorem to modular co-

of Artin

and of van der Waerden

finiteness criterion of E.

Noether

.... .

.

73

74

76

I

78

II

80

III

82

Appendix

Index

62

68

divisor chains

42.

variants

57

60

83

PART 1.

A new

I

Notation.

been found convenient in this book to introduce the signs which shall presently be explained. We say that two numbers

It has

II

and HI a and b are congruent modulo p if their difference is divisible byjo. This is commonly written in text-books on the theory of numbers as a

The

= sign

,

however, often

=

b

means

(modjp)

(11).

"is identically equal to,"

and confusion

we wish to use it with these two different meanings. When used as in (I'l) it gives no indication as to whether the congruence holds for all values of and b or only when a and b belong to a parwill arise if

it is

ticular field, e.g.

xp

is

congruent to x modulo p

if,

and only

if,

as is

an

We

shall therefore use HI to integer (a positive integer if p is even). to mean "is residually mean "is identically congruent to" and can only be used congruent to by Format's Theorem." Hence the sign II

||

in the cases

where Fermat's Theorem and

means

"

= means

"

II

ill

is is

equal to," identically equal to,"

means " is residually congruent if a is a positive integer, means "is

its

to,"

extensions hold. e.g.

2

=*=

e.g.

4

=f

x

^ 4,

e.g.

dP

identically congruent to," e.g.

6

,

II

a

mod p,

ill

3

mod 3, mod p.

a p -Hf a 2.

A

Thus

x ~ 4,

3,

Galois Fields and Fermat's Theorem.

A. Speiser* gives the following definition of a Galois Field. system of a finite number of elements forms a Galois Field if the

following conditions are satisfied (i)

:

The elements form a commutative group with

respect to the

addition law. (ii)

The elements with the exception

of zero form a commutative

group with respect to the multiplication law. (iii)

From

For any four elements the distributive law

be obtained, but *

it is

is valid.

the properties of a Galois Field can easy to see that the Galois imaginaries, hereafter

these three conditions

all

Theorie der Gruppen (second edition), p. 54.

MODULAR INVARIANTS and are therefore the elements

defined, satisfy these three conditions

of a Galois Field.

Let us consider a polynomial

where each of the

coefficients

0, 1, 2, ...,/>

where

!,

p

is

an

...

,

,

a

one of the following integers

is

a given prime,

the coefficients are the

i.e.

We may

modulo p.

of course suppose that &rt -HfOmodjp, and we shall further suppose that is irreducible modulo JP, i.e. that there exist no polynomials and s of degree less positive integral residues

/

^

< 2 Ml/ mod p. always possible to find a number b n such that b n a n \\\l ~l n + cn where + +