Modular Invariants

Cambridge Tracts in Mathematics and Mathematical Physics GENERAL EDITORS HARDY, M.A., CUNNINGHAM, G. H. E. F.R.S. M...

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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

/

^

than n such that

Now

it is

^

l

.

.

.

,

,

"

an + Cn-iO? 1 + if/ bnf 1

...

+c

is

called the normalised form of/;

and obviously

/

we

in 0, we are dealing with the case where can without lack of generality take the coefficient of x n congruent to 1. ill

ill

If/

ill

0,

0.

Since

any polynomial

/ and p

a residue

of order

n and it Each

and

(iii).

<

x with integer coefficients is congruent 1. Thus for any (r) we get for moduli

in

^n

to a polynomial of degree

<

These residues are the GALOIS IMAGINARIES

(#).

easily seen that they satisfy the conditions

is

(i), (ii)

some 6 (x\

(x) can be written congruent to

d can be chosen in p different ways, there are pn different n Hereafter we shall residues and so the ORDER of the Galois Field is p write this briefly as GF\_p' while we shall denote the field of all complex Since each

%

.

l

'\

numbers by CF. Fermat's well-known theorem states that

if y is a positive integer, then generalisation of this theorem for the case where y is a Galois imaginary is obtained as follows. Let u be a Galois imaginary

y

p

modp. The

\\\y

and not zero, then 8 ~* 8 u? then u if u 2 is true, then 1, w, w .

u~ 2 ur l u \ u\ u2 u?, ... are not all distinct hence t for which this 1. Let e be the least value of s ' ... u 6 1 are distinct. So are u 1} UiU, UiU 2 uu*~\ (

. .

,

,

II

II

:

,

,

.

,

,

,

u

e

~l

.

.

,

manner we see n that since there are p -l non-zero Galois imaginaries, e must be a e n q \. Since u divisor of q~p 1, and if y be any 1, therefore u Galois imaginary then y pn y. In Part I we shall write

where Ui

is

not one of

1, u, ...

,

.

Proceeding

II

in this

\\

||

J>*-15?, n

The p

n solutions of x* - x

||

P

n

n

(p -l)

are the

p

n

= d.

Galois imaginaries, therefore

MODULAR INVARIANTS n x"*>

-x

6

n

II

(a:

i^Q

- Ui\ where u = 0,

HI,

...,u q are the elements of

GF[p

n ].

Further generalisations of Perrnat's Theorem were given by E. H. Moore. For a first reading we would advise the reader to take n 1 through1 and d-p^p. The elements of GF\_p\ are the out, thus

qp

residues

modulo p

of the positive integers.

Transformations in the Galois Fields. In the work which follows we shall have to consider groups of homo-

3.

geneous linear transformations whose determinants do not vanish in the

The inverse transformations

field.

therefore exist.

We

G

shall call

the group where the coefficients belong to the OF, and we shall call r the group where the coefficients of the transformations belong to GF[p n '].

GI and I\ are the sub-groups of G and r respectively which consist only of transformations whose determinant is equal to unity in the field.

When necessary we shall denote the number of variables in the transformation by an upper suffix m, e.g. G m Gjn T m I\ w We obtain the order of the group r as follows* Consider the trans,

,

.

,

:

formations of the following type where the coefficients are elements of the GF[p*] t

There are p nm I possibilities for the right side of the first equation, for we cannot set HP-L = 0. If $ and T be two transformations which

x by the same linear function of the #*s, then /ST" 1 will leave unchanged and therefore will have a matrix of the following type

replace XL

:

"1

...

t)m,2

^m3

Such substitutions form a sub-group values there are fixed

(m

p

n

(

m ~^

of T:

vw and

if

we

possibilities for the a's.

give the fo's fixed if the a's are

Also

we have a sub-group of the sub-group the order of which - 1), where (m) means the order of the group T m thus ;

(m)

and using

this as

=p

n

(

m ~v (

p

-i)0(m- 1),

a reduction formula, we have

nm nm n nm 1) (p (m) = (p -p } (p -p*n ) *

Speiser, Theorie der

Gruppen (second

.

. .

(p

nm

1

-p***-

edition), p, 219.

)).

is

MODULAR INVARIANTS

4 4.

Let

Types of Concomitants. <(, ^) = ^n...i^i + n...i2^~ ^2+ 1

?'

be an m-axy /: ic which is homogeneous in the #'s. We shall call <#> the GROUND FORM. So far as is conveniently possible we shall reserve the letters a, 6,

c,

now the group

...

for the coefficients of the

of all

ground forms. Consider homogeneous linear transformations

Xi= Sa
(e

= l,2,...,w)...(4'l),

.7=1

subject to the condition that a v does not vanish in the field F. Let the shall use small old English |

|

matrix of the transformation (4'1) be H.

We

text throughout for the coefficients of the transformations. Now let 4>(a, x) become (a, x) under a transformation of the group; then the

be functions of the a's and the

a's will

Now

a's.

if

any function

C of

the a's and #'s exist such that

M

C(a, x)

is

equal to M(jSi) C(a, x) in the

field,

some function of the a's only, then C is said to be a CONCOMITANT. If C is a function of both the a's and the #'s, it is called a COVARIANT; if it is a function of the a's only, it is called an INVARIANT if it is a function of the #'s only, it is called a UNIVERSAL COVARIANT or an INVARIANT OF THE GROUP. These definitions can readily be extended to the cases where we have more than one ground form. We shall commonly use the term covariant to include invariants. The a's and the a's may belong either to CF 01 GF[p n \ so that we have the following types where

is

;

:

Type

the form p

and

both the

If

I.

a's

and the

a's

belong to

CF and

reductions of

are forbidden, the concomitants are then called ALGEBRAIC

HI

this is the classic type treated thoroughly elsewhere.

Type

If

II.

both the a's and the

a's

belong to

CF

but

p

ill

is

we speak about CONGRUENT concomitants. Miss Hazlett* this case as a special case of Type III.

allowed, then

has treated

If the a's belong to the CF and the a's belong to the then and a* n a are permitted. In this the reduction sp GF[p case we talk of FORMAL concomitants. These have been treated by

Type

III.

n

ill

||

"\,

Dicksont, Sanderson! and Hazlett*.

Type

IV.

If the a's belong to the

GF[p

n ]

and the

*

a's to

CF, then

Trans. Amer. Math. Soc. vol. 24, pp. 286-311 (1922). t Madison Colloquium Lectures and Trans. Amer. Math. Soc. vol. 15, pp. 497-503 (1914).

J Trans. Amer. Math. Soc. vol. 14, pp. 489-500 (1913).

MODULAR INVARIANTS

5

n

a are allowed. This type has not been treated so but we shall give to them the name of NON-FORMAL concomitants.

p

and a

ill

II

and the

If both the a's

Type V.

belong to the pn pW in 0, a a, E

a's

n

GF[p \ a. We

far,

then

three types of reductions are allowed, p shall call concomitants of this sort RESIDUAL concomitants, but in other papers on the subject they are termed modular concomitants. prefer to use is allowed. Type this term to cover all types where p has been II

II

We

V

ill

treated extensively by Dickson *.

concomitants of types II, III, IV and V MODULAR to them from Type I, the algebraic or projective case. By considering the reductions employed it is obvious that every algebraic

We

shall call

distinguish

also a congruent covariant

that every congruent covariant ; that every formal covariant is also a residual covariant. It is clear, however, that two formal covariants may be identical when considered as residual covariants or that a formal

covariant is

is

also a formal covariant

covariant

may be

ai*a 2

a L a}

When

this

is is

when regarded

zero

known

;

as a residual covariant, e.g.

to be a formal invariant

of/ = a l x l a2 ^ if JP = 2. we can use Fermat's -\-

considered as a residual covariant

Theorem on the

a's

and our invariant

is

aia^-0.

a^a^

We can extend our definitions to include what are known as cogredient Following Dickson we use the term point in the sense of homogeneous coordinates thus the point (y^ys) is identical with the point (ky\ky>Jsy*) and the point (0, 0, 0) is excluded. Now if the transforma-

points.

;

tions (4*1) of the variables #,,

...,

#m and the transformations

of the

coordinates ylt ...,y m have the same matrix H, then (y lt y m ) is called a COGREDIENT POINT. We shall now prove the theorem which will be . . .

,

GF

n required later, that all points whose coordinates belong to [p ] are such can transformed i.e. be into under I\, any any other point conjugate of transformations I\ by .

Now

(1, 0, 0, ..., 0) is

and the elements of

(1,

conjugate with

a2

changes of sign under

This proves the theorem, is conjugate with

in turn *

,

...,

OT )

_

(1,

aa

,

...,

a m ) under

can be rearranged with perhaps -

e.g. (2, 1,0) is

conjugate with

(1,

2,0) which

(1, 0, 0).

Amer. Journ. of Maths,

vol. 81, pp.

837-354 (1909) and other papers.

MODULAR INVARIANTS

6 5.

Systems and Finiteness.

We

shall call

a system of co variants 7TC

K

lt

,

...

,

-/Tff -i

a FULL SYSTEM

every other covariant / can be expressed in terms of these JTt 's. If the o- covariants 0) ..., a -i are linearly homogeneously independent if

K

K

every covariant / can be expressed as a linear homogeneous function of these JT/s, then we shall call the set JT ..., a -^ a FUNDA-

and

if

,

K

MENTAL SYSTEM. The reader should be careful to note that in some papers no distinction is made between a full system and a fundamental system. It is

is

obvious from the above definitions that every fundamental but that not every full system is a funda-

also a full system

system mental system. As a special case of a

FULL SYSTEM. The

s covariants

they form a full system; s covariants. These elements if (i)

system we have a SMALLEST form a smallest full system system exists with less than

full

N

N^... no full t

(ii)

B

N^ ..., N* form a BASIS of the smallest ^ form a basis of the full system or the system and K, ..., fundamental system as the case may be. The covariants of a ground form with respect to a group of transformations are said to possess the FINITENESS PROPERTY if there exists a finite

K

full

ff

full

system. given case, as

We

shall not

have to prove the finiteness property for any

we shall prove that it holds in every case with which we deal.

That algebraic covariants possess the finiteness property is well known. In another section of this book we shall give E. Noether's proof that modular covariants of all types possess the finiteness property. L. E. Dickson* also gave a proof of the finiteness of residual covariants. We shall consider these finiteness

proof till later.

theorems as proved, although we leave the

Most of the work done in the theory of modular covariants

has been concerned with the finding of a basis of a full system in some particular case. As we shall show, there exist several methods of residual invariants it obtaining covariants, but except in the case of is

very 6.

difficult to

say whether a given system

is

a

full

one or not.

Symbolical Notation.

One

of the greatest difficulties

for

many

years in the theory of

modular invariants was that no suitable symbolic method of treatment had been found. The method employed in the algebraic invariant theory obviously would not do, since it employed to a large extent multinomial coefficients which in certain cases might be congruent to zero modulo p. Let us take as an example the binary cubic. In the 8 as o^ where <* = algebraic case we represent it symbolically ,

*

Trans. Amer. Math. Soc. vol. 14, pp. 229-310 (1913).

MODULAR INVARIANTS

7

2 3 2 a/ = a/ a?! + Saj o^x-iX* + 804 cu, ^ X? + &/% 3 3 8 + a 2 3?2 mod 3. Thus for the modular case with p - 3, a/ does not represent the general cubic. The method employed in the modular case is merely a generalisation of this one, and it has the advantage that it can be used for the algebraic case also. In general then, we write the ground form as a

Now

Ill

a^

We

product of linear homogeneous symbolical factors*.

shall write for

example the general binary cubic as

/= ax* + bx^Xz + cx^x} + dxl ~ a 2 #2 )

(&#j +

2

#2 )

<*>

x

px yx

(y l Xl

+ y2 #2)

We shall note that any non-symbolic coefficient of a ground form when represented symbolically must be of the first degree in the a's, degree in the

first

/3'$

and so

on.

It is also symmetrical in the

symbols

Thus we have the important conditions which a function of

a, /?,

the non-symbolic coefficients satisfy It

(i)

must be symmetrical

:

in the symbols a,

(3,

Each term of the function must be of the same degree

(ii)

the symbols

We

in each of

a, fi, ....

have other conditions that this function be also a modular con-

comitant. These will be given in 9. shall use the small Greek letters

We

a,

/3,

.

.

.

,

8 for symbols.

symbolism no equivalent the difference in the two symbolisms

It should be noticed also that with this

symbols are required. To illustrate shall consider the discriminant of the binary quadratic.

we

f- axi + b

Let

If c

we represent /

= a* - ft and

2

2

(a/?)

we represent 2 2 (a) = & -4ac. If

The

as a x2

= /3x2 then a = a * = p 2

= 2ac - 1" =

f as

1

,

2

(4ac

is

,

&

= 2a 1 a2

).

% A, b = ajft + aa/2i> c = ogft and

a^ft, then a

invariant obtained

-b

l

2

the same in both cases, but in the former

= f An invariant need not, however, multiplied by a constant have the same form in both symbolisms.

case

it is

.

*

Sanderson, Tram. Amer. Math. Soc. vol. 14,

p.

496 (1913).

MODULAR ^VARIANTS

8 7.

Generators of Linear Transformations.

Capelli* proved that any linear transformations of m variables can be obtained by the successive application of a finite number of linear transformations of the following types :

It is easy to see that these in turn can always be generated

S^\

S

1

2

from

supplemented by transformations of the type X-L

=

Xj-x^

Xj,

%ic

x-k

(k

=1=

1,

k ^pf).

We have therefore the theorem Any homogeneous linear transforma:

tion whose determinant is not zero can be generated from transformations

of the following three types:

Type

11..

I.

.

1

.

.

.

.

1

.

... Type

II.

^=

rfc

i

(*=*=!)

Type

III.

x = Xj

e.g.

l

By a combination of variables thus

of matrices of

.

.

.

i

.

.

i

i

.

.

.

i

.

.

.

.

i

.

.

.

.

i

Type

III

:

*

1

Lezioni, p. 202.

for the case where

we can interchange any pair

MODULAR INVARIANTS by interchanging the

variables of

we get

II

Type

9 Capelli's type

S%\ and

by interchanging the variables of Type I we get Capelli's type Shtk "We shall give a short proof of the theorem in the matrix notation for .

.1 x

.1 x

.

1

and

fl

x

.

.1=

1

pfc 1

.

1

.

.

.

fc

.

.

.

1

.

.

1

.

.

1

fl

1

.1 x fl

1

.

= Pi

.

."] .

IT

.

.

..1

.

..1 Type

III

Now

1

where

ft

.

C

a 2 as

6i

t) 2

.

c

.

i

.

.

l

.

.

.

i

.

.

l

=

a i i

.

C2 C3

t) 3

.

tT 8 i

.

.'I

.

ai

.

1

.

.

..1

i

.

fe

we can obtain the type 1

and

ft

."]

1

Similarly by use of

.'

.

j"l

[".

.

i

.

Ci,

fci 2

= a x + a 2 tJ 22 + a s r2

,

fcw

= a2 tJ 23 + a3 cs

.

Now provided that the determinant Hi -Hf 0, we can solve these nine equations for the a 1? a 2 ... C8 This proves the theorem. The proof in the w-ary case is similar. |

,

,

|

.

from this theorem that any function which remains

It follows

in-

variant under every transformation of the above three types is invariant under every transformation of the group. Also if the group is r then n Thus a formal the fe in Type II must be an element of the ].

GF[p

covariant

is

the 6r.F|j?

invariant under

n ],

unless

it

Type

II

if,

and only

if, ft

be also a congruent covariant.

is

an element of

MODULAR INVARIANTS

10 8.

Weight and Isobarism.

Following the method of Elliott* that our ground form is an w-ary

and

define the term weight. Suppose

l-ic

in

m

the

variables #1

-

.

.

,

,

xm

,

the coefficients of

let

a?j

we

,

an

2

,

. . .

,

1

# m _j

#i~~

,

x^

have the

... ,

suffix

0,

2,

J~

T ^l^m

is,

l~ l r r ^2'^m

have the

#j^,

that

l

"'

'

suffix

l

7-1 *

"

'

'

*

/,

the suffix of any coefficient

is

equal to the power of xm which

multiplies in the form. In addition we say that the suffix of x m is 0, while the suffix of x t is 1 for i=tm. The WEIGHT of any term is defined

it

as being the sum of the suffixes of its various factors. Thus the weight of each term in the ground form is /. If a polynomial in the coefficients

a and the

variables

x be such that each term

then the polynomial "

isobaric

"

is

said to be ISOBARIC.

according to Elliott, but

we note

is

of the

This

is

same weight

shall therefore

make

9

that to state quite clearly

what we mean we must use the phase "isobaric with respect

We

w

a definition of

the following definition

:

to

a m ."

If a polynomial is

isobaric with respect to all variables, it is said to be

COMPLETELY

ISOBARIC.

For example suppose that /= a^x^ + a^x^ + a z x be the ground form then g = a^x\ + a s ^2 + ^2^3 is isobaric with respect to XL but is not isobaric with respect to ^ 2 or #3 and is therefore not completely isobaric. By considering transformations of Type III we see that if a concomitant :

isobaric with respect to any variable, g could not be a covariant of/. is

9.

it is

completely isobaric. Thus

Congruent Concomitants.

By definition it is clear that congruent concomitants only differ from the algebraic concomitants in that it is permitted in the former case to make the modular reduction p 0. Let us take as our ground form \\\

<

= CL,./^, *

...

,

83.,

where a^ = a^j

Algebra of Quantics

4- ...

(first edition), p. 38.

MODULAR INVARIANTS

11

Now if C be a congruent a, ft, ..., 8. a function of variables x and the non-symthen we can write

and where there are r symbols co variant of

where

bolical coefficients a,

C

is

C(a, x]

where Jf

As

M (a)

III

(7 (a, #),

a function of the coefficients of the transformation only. in the algebraic case* we can shew that C is homogeneous in the is

variables #, or else

is

the

sum

of covariants which are

in the variables x.

We

C

in the variables x.

is

homogeneous

Now

homogeneous

can assume then without loss of generality that

considering transformations whose matrices are of the type

\\

.

.

ft

.

\

*

a non-zero scalar, we have easily that C must be homogeneous in the non- symbolical coefficients a. By considering transformations of the type

where

fc is

we see that must be symmetrical with respect to the suffixes both of the #'s and of the symbols a, /3, ... 8. Let us represent then our covariant C (a, x) symbolically. single ,

A

term of this representation can be written as follows

where

jS=S! r Jfi

and

+ ss +

...

=A + h + 1

+ sm

+h m -k

>

2

:

^

l

...

= Jj

m

... l

are constants for every term c { of C.

Now

if

#1

fc#i

,

and

then

so that

Now

cf tt

T

= fc T

i,

where

i

= ltai,

T= ^ + ^ +

. . .

+ /i -

i

.

must be the same for every term ci} so that since ft is an we have that T is a constant and therefore

arbitrary non-zero scalar

*

Elliott, loc. cit. p. 40.

MODTJLAB INVARIANTS

12

T+ S C

T+ 8 is the

a constant, but

is

weight with respect to

x and

isobaric with respect to

a?i,

so that

completely isobaric. This gives us the theorem congruent concomitant is completely isobaric. Conversely if a formal concomitant is isobaric, then T + 8 is is

:

a constant and so values of

It

and

is

is

T,

and

therefore

l

is

A

so the concomitant is invariant for all *

Miss Hazlett

therefore a congruent concomitant.

proved the converse for the binary case. We shall also extend to the If a congruent con7-ary case her proof of the following theorem t :

comitant has factors, these also are isobaric. For, let C have factors CL and Cg which are not both isobaric, then the term which has the

6y2

is the product of the terms which have the greatest/least weight in Cl and (72 respectively. The theorem follows at once. By considering transformations whose matrices are

greatest/least weight in the product

1

x

ft

fe

we

find as in the algebraic case that the following relations hold for

congruent concomitants

:

(91),

K> the order $, the degree ff&nd the weight is W$ with respect to any variable # and where r is the degree of the ground form.

where the index

is

t

Relation between Congruent and Algebraic Covariants. z It is well known that b - ac is an Let/ = axi + 2b&i&2 + c

10.

-

invariant of this form.

Also (b 2 if

p=

3,

It

is

algebraic therefore also a congruent invariant.

acf must be both an algebraic and a congruent

invariant, but

then a

(6

-ac)

8

6

ill6

-aV

and so b* - a s c* must be a congruent invariant. It is algebraic invariant. Thus it would seem that for one ant ( 2 - ac) 3 we have four congruent invariants, viz.

these are

all,

not, however,

an

algebraic invari-

however, congruent to each other, and therefore represent *

Trans. Amer. Math. Soc. vol. 24, p. 296 (1922).

f Ibid.

p. 297.

MODULAR INVARIANTS

13

the same congruent invariant. It will be more convenient therefore if we neglect any term in a congruent co variant having the factor p. If we proceed in this way we infer that one, and only one, congruent covariant is obtained from any algebraic covariant, and we can therefore represent such a covariant symbolically since every algebraic covariant can be represented symbolically. now put the important question. Can every congruent covariant be represented symbolically? In other

We

words, does there correspond to every congruent covariant an algebraic covariant? An answer to this question for the general case has not yet

been given, but clearly

forms the keystone of the symbolical theory of symbolical theory is obviously not of much use

it

A

modular covariants.

covariants exist which cannot be represented symbolically. It is extremely likely that every congruent covariant can be represented symbolically, but in the absence of proof we shall have to divide

if

congruent covariants into two

sorts,

and non-symbolical congruent

covariants,

symbolical congruent covariants i.e. those which cannot be

represented symbolically. If it be proved later that this second kind does not exist, then our discussion of the first kind will be applicable to all congruent covariants. Let C be a congruent covariant: then will

in

Jf(7, where Mis a function a

Let

of the coefficients.

_

similarly

be a sum of terms

C= 2 Pi= S MiP

thus

C= 2

P* and

say, so that, for all values of the a's,

it

i=l

S Jf
Mt\\\M is

2 MP,,

ifi>a

Mi

and

M

ill

i=l

then must

prove that

iy

b

i-l

We

P

i=l

_

a

C

ill

ifi^a.

congruent to a power of

|iE|

by the same argument

as that used in the algebraic case. The proof of the binary algebraic case given by Grace and Young* can be extended to the w-ary case.

Thus each Mi

= (a*

1,

...,&) is

a power of the determinant of the trans-

formation or else zero and

is

cannot be applied here.

Miss Hazlettt proved this for the binary

therefore isobaric since Fermat's

Theorem

case.

Miss Hazlett and R. Weitzenbock have proved in special cases that every congruent covariant can be represented symbolically. *

Algebra of Invariants, p. 22. t Trans. Amer. Math. Soc. vol. 24, p. 297 (1922).

MODULAR INVARIANTS

14

R. Weitzenbock's proof

Let

C be

is

as follows

:

a congruent covariant, then as before

O(a,^\i\M(a)C(a

C (a,

or

x)

=

|a|*

t

ai)

C (a,

and

x) +

M

s

(a)

III

|a|

,

pD (a, x, a).

8 = cs a operate 5 times with the Cay ley operator O. Then O*|l3| constant so that &*C(a, x) is independent of a and is therefore* an

Now

QD

(a, x, a) must also be independent algebraic covariant K, say. Now this call we shall and of a expression E. Then

K= Now

if c s is

c.

C (a, x) + pE

not divisible by p, then

covariant and

is

or

K\\\ c s

C (a,

C (a, x).

K ,r)

ill

,

which

therefore representable symbolically

an algebraic

also

.

C (a, x) very often

so that

;

cs

representable symbolically. Unfortunately divisible by p, in which case the proof does not hold.

is

is

GS

is

In fact t

\m "JO

and

is

therefore divisible

by p unless m + s-1
i.e.

unless

Miss HazlettJ proved as in the theory of algebraic semin variants that of a system S of binary forms, which is of degree g and weight w then /is congruent, modulo p, to a product of a if

/ be a congruent seminvariant t

power of a

and a symmetric polynomial

P in symbolic ratios a Pi Moreover P expressible as -

.

,

,

.

.,

i

which

is

homogeneous in these

ratios.

is

a

this she proved polynomial in the differences of these ratios. From that every congruent invariant of a system of binary forms is congruent

an algebraic invariant. The symbolical representation of such invariants follows immediately. Since symbolical congruent covariants are congruent to algebraic covariants, they can be represented by the corresponding symbolical

to

covariants follows expressions. The fmiteness of symbolical congruent at once from the algebraic case. It is seen that such symbolical congruent covariants are very similar to the algebraic case, and we obtain full

systems of the one from the *

full

systems of the other.

Weitzenbdck, Invariantentheorie, p. 147.

f Weitzenbock, loc. eit. p. 16. } Trans. Amer. Math. Soc. vol. 24,

p.

298 (1922).

Trans. Amer. Math. Soc. vol. 30, p. 855 (1928).

MODULAR INVARIANTS 11.

15

Formal Covariants.

Not every formal covariant is

isobaric; and so while every congruent covariant is also a formal covariant the converse is not true. As in 9, r must take the same value for tt every term ct of C (a, sc\ but according

to definition ft is no longer an arbitrary non-zero scalar but is now an element of GJ?[p n and we may therefore reduce {t r by Fermat's Theorem. Thus it is no longer necessary for T to be equal to a constant but jTmust be congruent to a constant modulo q for every term c% It ~\

.

follows that the weights with respect to all variables must be congruent modulo q to each other. Thus the weights of the different terms of a

formal covariant must be congruent modulo q. In this case we say that a formal covariant which is not isobaric is PSEUDO-ISOBARIC. The uni-

we shall now discuss are all pseudo-isobaric, and since they are independent of the coefficients of the form they are also residual co variants, without any reduction being performed. versal co variants which

The equations' (91) must hold also for formal concomitants replace the equal sign by a congruence modulo q

if

we

:

rH

thus

rH+ (m 12.

Let

$

in

mK

i)8\\\m Wi

(mod

<

(mod

<

Universal Covariants*. X-L

.

,

.

xm be a set of variables which undergo the

.

,

formation

following trans-

:

TO

(e

where each

% belongs to

Gl?[p

xt = 2

then

j=i

or

a?f

i>

H

,7=1

n ]

=

l,

...,

m\

and where

(KMT +P n (--^

a# #f and thus #f"

..

'

j-l

We

have therefore the important result that the set #f"> modularly cogredient with the set #1, ..., #m where t is any positive .

,

integer.

Now

let

[>!,

us write ..., e

m] =

Dickson, Trans. Anier. Math. Soc. vol. 12, p. 75 (1911).

. .

,

MODULAR INVARIANTS

16

As

must be an invariant

in the case of algebraic invariants this

since

the rows are cogredient. [ely ..., em is an example of a universal covariant ; it is frequently referred to also as an invariant of the group

all

~]

is independent of any ground form. There is no expression in the theory of algebraic invariants corresponding to this, if there are less

since it

than

m

sets of cogredient variables. In particular we shall write

[m,m-l,

...,

g+1, 8-1,

We have seen in 4 n GF[p ] are conjugate

all

under

I\.

4r> =

Iff,

TO

,

7==g.,.. J^m

points with coordinates belonging to Thus if a covariant vanishes when the

one of these points,

#'s take the values of if

that

1,0]=

...,

it

must vanish

for all

;

therefore

a covariant contains one factor of

Em = n n where d/p n denotes that each d in the product takes all the p n possible values, it will contain every factor and therefore m itself. NowZm has the factor xm and therefore the factor 12m> and comparing coefficients we have

E

that

Lm \\\Em

Similarly every Lm m as a factor; hence

L

the index of every obviously a factor of

.

m and rational and

has the factor

Q mt8 is Lm is 1.

Em

ac

Also

is

therefore either zero or has

is

an absolute covariant since

L m -i

divides

Lm

since

J m -i

'is

.

Consider the following determinant which vanishes identically.

^n(m-i)

pn(m-i)

m+

1

rows

m -2

rows

...

i

m cols.

i

-1

cols.

MODULAR INVARIANTS By

we get

Laplace's development

W-2

(-1) + (- 1)"*- 1

K

m

+ (-

If therefore 1

..

.,* +

[m-1,

l)

<

1][>-1, ...,*+l,*-l,

...,

O,

,9

<

m

!,

...,

- 1

s-l,

0] [m,

s-f

...,

1,0]

...,

0][>-1, .,

...,

s-

1,

1]

1, ..., 1]

=

0.

we have

Qn-^L^ - Q L m 2_

=l

17

vl , t

+ Lm

x

(Q^^L^T,

andso For the case where

=

.s

we

1

find that the exponents in the last row of 2n If s = 1 the exponents of the p

the determinants are equal to

(m +

2)th row are

L

nl

j9

w (m ~ 2)

Qm-i,

so that

i

m

.

and we have

An-i

Qw,

i

L m Lm _

l

+

+

Qm.i^Qm-i.il-r^-}

/

+

1

Expanding

L M by the last

l

= 0,

L pm _\ p

(12'2),

\p n -l

T

^- =(^)

so that

Lm Lm _

C-i,-

(12-3).

2

column we have

m~ 2

na

n(Ttt-l)

(12-4).

Thus

Lm is

divisible

by L m ^

and each expression

(12'1), (12'2), (12*3)

can be given in an integral form.

The degree

of

Qm)8

is

p

nm

ns p and

pmm-i) + 13.

fff

that of

+pn +

Lm

is

i

(12'5).

Dickson's Theorem.

Dickson* proved the fmiteness theorem for universal covariants and gave a full system. The proof of this theorem is of course included in E. Noether's Theorem (v. 43), but we require Dickson's proof however in order to obtain the full system. The theorem is stated as follows :

The functions L^ Q M i, ., system of the invariants of 1>. ,

QM

,

M -i are

independent and form a full

* Trans. Amer. Math. Soc. vol. 12, pp. 75-98 (1911).

MODULAR INVARIANTS

18

ft

The theorem < m where m

We

proved by induction.

is

and prove that

2,

assume that it is true for = m + 1. Let / be /x

true for

it is

m+1 The coefficients any homogeneous integral universal co variant of T w of the various powers in xm+l in / must be universal covariants of r and hence by hypothesis are integral functions of Lm ... Qm m -i' We .

,

,

,

,

suppose then that

/= / I

where each

t

is

+ a m+1 /i+...+ #4+i Ir

an integral function of

L mt

.

.

.

,

Qm> m _j

,

.

Then /

is

an

aggregate of terms

_ L -c m '

/' t

Consider those terms in which is

a minimum

b,

and

/l&l'

JTfl'

ty

a

a' is

f)b'm-l im _ r

C4

...

/M

minimum

so on, so that finally

the sub-set where

a,

6'

we obtain the unique term

Expanding by (12*1), ..., (12 '5), we see that the terms of minimum degree in x m are included in

<

where

,

m 1 = c i^-i n C-i s=2

t,

minimum

Similarly the terms of the

where

and

this t.

except

c

= c^< _ 1

term T

is

1

...

(a <

will only occur

once in

(a,

= ,#" + jJ-

=a t

i

.1

+

^

and not

^in

TI

where TI =

c

)ft <

rf):

any other product

XT~'-

<-i
m a term of a universal covariant of T

r

1

the isolated term T and therefore also

a term of a universal covariant of T m

versal covariant of

0-

contains a term

a??-

Now /has

Therefore r x

t

*i

degree in x m -.i are included in

L^_n Qi"*".., s=3

1

is

=

so on, so that finally

r

and

,

(i = P" +

,_i

2 ,

therefore

GX*

is

~l

and

.

cx-f is

Similarly

a term of a uni-

a kQ^ whence a =

a term of

ad.

w

the degree of Qm 8 is a multiple ofjt> and since a is a multiple n n This is therefore true of p therefore the degree of t is a multiple of p

By (12*5)

,

,

.

,

of

and since the degree of

t',

of p

n

p to

;

Lm

is a multiple p a multiple of g, for apply a transformation of determinant then the Q's being absolutely invariant we have pa

is

prime to

therefore a'

it is also

/

'

,

II

MODULAB INVARIANTS hence is

a'

in

a mod

also divisible

From

(12-1)

19

is divisible by d and therefore by g, so that a' Hence in every term t' of 70> is a multiple of d. we have

but a

'

d.

by

(12'4)

and

,.

=

ri..-i +

<^-i

(the factor Q, _ 1

if s

-

-!..+

Gl)

m - 1).

=

8 being suppressed In these series the exponents of xm differ by multiples of q. Let b 8 =pPsJE}8) where B is prime to p. To obtain the p^tli power of a sum in a field having modulus p we have only to multiply every exponent ft

'

by pP. Hence

m,l

^qp* re nP^ m-l +f^m L m~\ V m -l,l + dbi

=

1

r>

_ np

/)&

y m ,8-ym-l

n X~,qpP* +B * m ,

t

9-l

o.^ LT m-\

e

where in the

m

where

o-

>

last series s

Hence

1.

1,

and

the final product

Q s

/c

>

1

and the

'

' '

......

(13"2),

>

= db 1 -qpP*-

8

r\jP*

f\a

^ m-l, s + p ~pn+ ^ ...... (13'3), ,

Hf-l, s-l

where e,= b s

if s

(13-1),

.

where n bs

...... 1

factor

n

^OT _i,

8

is

to be suppressed

contains the terms

TO _i, a is to be suppressed has the values 2, ... o- ,

if

o-

1, o-

= m-~l, and where +

1,

...

,

m-

1

in

.

A

< w, then qp^ < d. The product t contains but one term First let with the same set of exponents as T. For if we employ a term of (13*2) If we employ after the second, the exponent of xm exceeds that in T^ .

the second term in (13'2) we must use the first terms in (131) and (13*3) and hence get TI itself. If we employ the first term of (13*2) we must use n the first term of (131), and obtain ap +db! as the exponent of Lm,^ in n

the product of the two, which is greater than e+ap which is impossible. Suppose that 7\ is a term of a product t' distinct from t. If a' > a, then a'

^ a 4- d,

ponent

a'

since a'

of

xm

in

and a are multiples of d. Hence the minimum exwould exceed the exponent of xm in Tlt It follows

t'

MODULAR INVARIANTS

20

that a' = a, further by (11 '5) and by the homogeneity of the universal co variant,

m

-pns Hence (&/ - b^)p is a multiple of p Thus in bi = pPi BI we have, since n

zn ,

)

=

so that bi

............... (13-4).

h mod jp w when

ill

a = a.

& and /?/ are both less than n, &'=& ........................... (13-5).

I\ cannot therefore occur in terms of

other than

t'

'" ...... (13-6).

If we employ the second term in the parenthesis we must take the term of each Qm>ii free of xm Then &/ = 61, from the exponents of L m ~i, and 6/ = 6 S (,s - 2, m - 1), from the exponents of But this is -i .

. . .

impossible, for (13*6)

t'

we obtain

Q-i,

,

*

we employ the

If

.

term in the parenthesis in in the product of the first two

first

L

as the exponent of m ,i since bi = when a' = a.

n n ap + dbi >e + ap

Hence our assumph have now shown that Ji occurs as an isolated term of the invariant. But, the exponent of x m is not a multiple of d, and the factors

tion

We

is false.

coefficient

B

,

l

is

not

Of the numbers

Hence the

zero. n

fore bi is a multiple of p b 2)

...

case

&
is

excluded and there-

.

,

b m _i not multiples of p

n

be the one with

let b a

A

T

term of t with the same set of exponents as a the smallest subscript. can be obtained only by taking the first term of (131), (13*2), (13*3) for s < o; for otherwise the exponent of xm is divisible by d. If we use the second term of (13'3) for s = cr, we must use the first term of (13*3) for > o- and then obtain T If we use the first term of (13*3) for s = cr the

s

ff

exponent of Q m -i Next if e in ff

t

T

bi

is

.

ff

-i

.

in the product is

Ta occurs in

a multiple of p

n .

t'

p

n

b ff

,

which exceeds

distinct from

t,

then

a'

-

a.

its

exponent

From

(13 '5)

Analogous to (13'2)

Hence we take the first terms of (131) and (13*7). In the product of these n n It follows that two, the exponent of Zm _! is ap + dbi > h a if bi > bi +p n bi ~bi. Ifo->2 bz is by hypothesis a multiple ofp and so is b z by (1 3'4), .

f

}

and we must take therefore the first term

Q ^^

does not occur in the expansion of Qmi8 for the exponent p n b s we conclude that b 2 =

s_1

s>

'

,

show that we must take the

first

b>2

term of Q*

.

of

Q

*' .

2

Since

2 but occurs in

Qm~

1} i

Ta with

In this manner we can (s

<
'

bs

= bs

MODULAR INVARIANTS (s

=

.

2,

.

.

,

or

- 1). Then

by (13'4) bj

ill

ba

21

mod p n whence

ft/

= ft,.

If

we

b

we must use the first term in Q^ g (s> o-) Q and we obtain T if, and only if, b s ~b 8 (s = am- 1), as shown by the of but in this case t' = t. If we ^ 8 comparing exponents Qm-i, (s o-), employ the second term in

ff ,

'

. .

ff

employ the first term in $!%> the X^
CT

definition of

We

.

9

total in

T

ff

,

exponent of since &,/

^

Qm -^

a -i in

t'

is

b a in view of our

.

Ta occurs as an isolated term of the inBut the exponent of xm is not a multiple of d since b a is not a multiple of p n and the coefficient B y is not zero in the field. Hence have now shown that

variant.

,

our assumption concerning b = n ofp*. Put b, p c s then

ff

is false,

so that b it

. . . ,

b m -i are all multiples

,

an invariant of F m+] in which /
is

.

function of

As a

Lm+lt Qm +i,

t

(i

invariant with coefficients

For

if

l,

...,

).

Dickson* has proved that any integral 2 n in the GF[p } of the group T is an integral

with coefficients in the GF\j) n have still to show that these universal covariants are independent. they are not, then there will be a relation

function of

We

=

basis of the induction

QZ1 andZ

2

-^m+l^l C-^m+i? N^m+l,

'\.

15

>

tym+l, m)

~*~

-t*

CM>W+I,

1)

j

tym+i,

m)

......

= Putting %m +i

and substituting

for the $'s in ^?

~

(13-8)

we have

B=

but I/m and the Q,n g are independent by hypothesis and so 0, so this factor and that the relation (13*8) has the factor m+i Removing = and the repeating the above process we prove every successive covariants m+lt Qm+ i,i, ..., Qm+1 m independent. ,

L

.

B

L

As a

,

basis of the induction proof used above

AL

we

notice that there

+ CQ\ i = between the universal covariants of P, by putting #2 = we have (7=0 and finally A = 0. no relation

2

is

for

This concludes the proof of the important theorem. It seems almost certain that a similar theorem would hold for the *

Trans. Amer. Math. Soc. vol. 12, p. 4 (1911).

MODULAB INVARIANTS

22

case of several cogredient variables x, y, z, ... and that a full system of universal covariants in such a case would contain determinants and

quotients of determinants in which only the following rows would appear

etc

:

-

of'

The proof of such a theorem for the perfectly general case by the above method would be cumbersome in the extreme. W. C. Krathwohl* has proved a theorem of this sort for the binary case where there were two sets of variables. The full system which he gave is, with slight changes of notation, as follows

:

xf

yf

x*

y\

x?

xl

yf 2/2

xf 2/2

2/2

#.= 14.

Formal Invariants of the Linear Form.

Although the theorem of the previous paragraph is strictly concerned with universal covariants only, it at once furnishes us with a method for am obtaining a large number of formal invariants and covariants. If % .

,

be any set of quantities which M

is

contragredient to x ly

. . .

,

xm then ,

.

.

,

so also

n

a^ contragredient and we at once obtain the following of invariants the linear ground form a 1 x1 + ... + a m xm> pure is

the set a^

.

,

. .

;

,

L(a)m>

where

L(a}m

is

obtained from

we can form covariant.

Q()m

obtained from

Q m K by ,

t

*

Lm

(5=1, .",^-1)

by substituting a for

the same substitution.

As

(U'l),

and

x, Q( a m}S is in the algebraic case

inner and outer products and each of these

)

must be a formal

We shall presently exhibit a system of covariants of the formal

linear binary form *

modulo

3,

but we shall

first

consider

Amer. Journ. of Maths, vol. 36, pp. 449-460 (1914). f Bulletin Amer. Math. Soc. vol. 21, p. 173 (1914-15).

how Glenn t

MODULAR INVARIANTS He

found this system. n

x

\

. . .

,

was the

x^

. . .

,

,

xm

Since

,

We shall call it the MODULAR POLAR.

an invariant operator.

is

use modular operators.

first to

are cogredient with xt

23

Similarly

we have the MODULAR ARONHOLD OPERATOR pnt

9

if

6

6

,

,

9

pnt

*

9%

w(t).

da^"

**

we have two m-ary quantics whose coefficients are a a^and M respectively, then we have two modular Aronhold operators

Again, ...

9

pnt l

da

..

,

VMl

9

nfMt

9

9

<

Glenn also defines MODULAR TRANSVECTANTS

m

TV (ft.

,

t\

for the binary case.

For the

^

functions <j6 1} 2 m and if the degree of general case if we have ... sm is obtained is ^-, then the rth transvectant with respect to s* 8 S <

<

. . .

,

,

t

,

,

by operating r times with the Cayley operator O on

dividing by

x?* m

for Zi.

By means

-*=^ rr^ \ti-r\ti-r ,

We

.

.

~^~.

-r

,

and then substituting #?

?

*

2

for

yit

...

,

shall write the expression thus obtained as

of such

modular operators Glenn gives the following system form for p - 3. He does

for the formal co variants of the binary linear

not show, however, that it is a fundamental or a the symbolical representation

3

(a a),

full

system.

We append

MODULAR INVARIANTS

24

Proceeding in this manner we can easily find a great many formal covariants, but so far we cannot say whether every covariant can be

L

expressed in such a symbolical form. We do know, however, that mi variables, Qm,i9 - Qm,m-i form a full system of universal covariants of

m

)

so that interchanging as for x8 we have the fmiteness of formal invariants modulo p of the w-ary linear form

a 1 a?1

We

+...+a m afm

.

can write this basis as L(a.)m> Q(a)m,l,

>>

Q( a m, m-1 ),

If the generalisation of Dickson's Theorem for several cogredient variables were known, then of course we should have a full system of formal

invariants of a system of linear forms. Utilising Krathwohl's Theorem we have the following full system of formal invariants of a pair of

binary linear forms

"1 &1

15.

The use

We

noticed in

of Symbolical Operators. 14 that the modular Aronhold operators were

symbolical operators.

The symbolical

9

&

V

riot

operators ,,

9

/

5 \

A

are also invariant operators if used with certain restrictions. function has no meaning unless it is symmetrical with respect in the a's, fi's, . . .

to the

and of the same degree

. . .

a's, /?'s,

in each.

Thus the symbolic

operators can be used provided the expression remains symmetrical in the a's, /3's, ... and the total change of degree is the same for every

symbol

a, /?, ....

aS 1

(

Thus

2

(a/5)

is

an invariant of/= a^,

p = 3 whence :

1

Thus we get the new invariant (as a) (/3 3/3) since (a/?) 4 is an invariant. Great care must be employed in using these operators to ensure that the factor

p

does not arise through their use.

This factor

may

be

MODULAR INVARIANTS

25

may not. We shall show an example where it is not trivial. For further use we shall give some symbolical formal invariants of the

trivial or it

binary quadratic

p=2

:

/= a^x* (a) 2

(a

3

/?)

2

(/3 /2)

4

+

a 1?

HI

m a?

a)' (a a) 2

4

(a a)

(aa 2

2

(

(/2 /3)/(a

(

2

2

a)

III

(/3 /5)

That symbolical representation of covariants can be very complicated seen by the following example of a covariant of/:

is

Xi

a,!

(a

comparatively easy to write down many symbolical formal is not always easy to find what their non-symbolical representations are. A good insight into the structure of such symbolical It is

covariants, but it

is obtained by considering a few examples. To lessen the work entailed we can use the following method of abbreviation for the

covariants

binary case

We

:

a i Sa 22/V/?2 as

write

aiV& & ~ 8

1

V&V

+

iVA/V

as

V

aS

20jV&

3

4

1

2

1

-

(3

I

3

5

I

5

5 ;

1)

(TQ

;

+2

I

[

1

3

I

1

5 J

5| 5|

|

5 |

a lt /3 l} y with strokes is, we simply write the indices of etc., between. The coefficients remain full size while the indices remain their own size. The figures in the oblongs at the side show how many As an example of this notation we a's, /3's, etc. appear in each term. shall show that it is not in general permissible to divide by the modulus, even though the modulus appears only as a factor due to an operator. Let that

l

/= a

Q

Xi

+ diSCiM* + a%x>2 =

%

= a1

2

Then

(a/3)

2

-2a

1 a.2

a x /3x

and

. . .

,

p=

,

3.

A/32

Now 1

=

6

[V&W - <&

&

1)

K)

(

1

-a.W + o.WA = 6 [5

I

(3

-

1)

+

4

|

(-

0)

+

3

|

5

+

2

|

+

1

|

(~

5)

+

J

(- 4

+

2)]

......

]

5

5 J

|

[

|

(151)

MODULAB INVARIANTS

26 this last

must be a formal invariant modulo

to divide

by 3 6

here, then |

-

(8

be an invariant,

+

1)

4

|

{-

Call this

Now

+

0)

8

+

5

|

2

+

|

1

(-

I

+

5)

s

is 3

-

o

|

1

4

o

1

;

(a/3 ) is 3 2

= =5

3 2

so that (aV) (a/F)

2

-

)

|

o

i|o-o|

IS 2

(a/3 )

3

|

2|o +

+

4

3 2

(p*p) (/3a

3

I

)

- 2

o

divide this by 5|(-3 + l)

|6|(-3

+

3 I

=

=

L K+j-

:yT

Now

=

4

|

3

(a a)

is

-

1

(-4 +

2

o

1

|

e

2

+

2

(-3) f

|

-

o

2

;

|

|

I

(-

|

+

s)

1 (-6)

[7[7|

;

be an invariant and

1

(-6)4-0

|

1

1 ;

J6|

.

(-5 +

|

6

6

3);

|

|

|.

1

[

1

1

2)

2|

(+5 + 8-1)

2

(4-2)+l (3-1);

+3 (-3 + |

(s

I

|

o

-

1

1

1)

o) (o

(3-1) +

so is

+

1

we may not divide by

that the operator

I

a

|

J

[

|

(- 5H 3 + 1) 1 (-

|

1

3

-

o

(-3 +

.

6

1

|

(-

6-f 3) '

3

(4-2)

|

|

|

...

^^ I

5

I

5

I

.

1);

1);

4

4

I

a + a 2) but a + a 2

under the transformation of matrix

s

- 4 + 2)

2

3|(4-2-0)

(^) =

an invariant

variant; hence

|

|e; o)

1

6

3|(6 + 4-2)

=3

K

(6

o

+ 3|<6-4+2-0)

4!(5-3-l)

5|(-4 + 2 + 0)4|(-5-3 + l)

if

1

1

o

+

s

1

3 |

say, will

,

(a/3)

+ 4|3

5|(4-2)

Thus

J

|

+ 4|(-4 + 2 + 0)+8|{-5-3+l) + 2|(4-2-0) + l|(5 + 3~l) +0|(4-2)

+ l) + 5|0

SO that

|

[

|

|i

3;

+

i |3

+

We

5

5

2) j

o

|

8

(a a) (a/3

<- 4 +

|

K.

(a a)

Hence

were permissible

If it

3.

would

is

Therefore

.

J

I

not invariant

K

is

not an in-

3 in equation (151). This shows

8 operating on a

is

an invariant operator only

in virtue of the factor 3 produced.

16. Annihilators of

Formal Invariants*,

In this paragraph we shall consider a method utilised by Glenn and Dickson for finding modular invariants in special cases. The modular *

Amer. Journ. of Maths,

vol. 37, pi

75 (1915).

MODULAR INVARIANTS

27

differs somewhat from the algebraic counterpart. The theory he best illustrated by a simple example. We wish to find formal invariants of degree 4 of the binary quadratic form modulo 3. Let

annihilate

will

fand

x

let

Then

/=

where

Let

(a aia>2) be (}

an invariant and

be written

let

<

ai

,

then

+ + i

[

II >

2

]

+

Q 8 ,

in

+

(toi

+

2

i

4

o)

*

4

]

..................... (16*1),

which we need not go farther than the fourth derivatives since the

invariant

only of the fourth degree. can write equation (16*1) as

is

Now we

<

(tf

i#2 )

~

(oi2) = t$i + t

Equations (11*1) show that <

(a)

<

2

S2 <^>

+

... 4-

must be of the form

= A^af + APafa* + A^afa? + A^af

where the A's are undetermined constants. Hence

We notice also that whenever the modulus 3 appears in the denominator, it will also occur in the numerator. We are still at liberty to cancel out

MODULAR INVARIANTS

28

as we have not yet introduced into our argument the fact that a modulus. "We have so far proceeded exactly as in the algebraic case except for the fact that in the algebraic case the invariants are isobaric and hence we would have to set several of the A's equal to zero.

these

3's

there

is

In the formal case modulo invariants

if

we put 8 + + A^ - A,^ + A^ T

A^ A^

ni

ill

and it is

8

3, t

88

8B

t

II

+

7

5 II

t

II

t

and hence we obtain formal

,

= 0. This gives the following relations

87

AP Af\ A^ + AJ + A ^ - AJ 0,

III

0,

in

0,

must be invariant under the substitution X = x 2 invariant if we interchange a with a 2 and - #a with a lt

since

<

L

A^ A^ (a)\\\A j

Hence we have two binary quadratic,

2

(4)

2

<^

(

+ a Q a* a 2 -

/ or J

This method

is

III

3

#2

,

x%

=

x\

,

so that

Aj.

-

o

#/)

linearly independent invariants of degree 4 of the

mod

3,

Glenn points out that I+J\\\ that either

A-j

0,

ni

III

Hence

:

ni

D

2

where

D

is

the discriminant of/, so

reducible.

is

somewhat cumbersome but

a different method Glenn found a

it

leads to all invariants.

system of the binary cubic the binary quadratic mod 3. His method is somewhat long and will not be included here, but the procedure will be indicated. As in

By mod 2 and

full

the algebraic theory, the coefficient of the highest power of Xi is called the leader and the leader must be a seminvariant. We can always

L

remove

factors so that a covariant can be taken generally as having the highest power of xl equal to the order of the covariant. If we have a fundamental system of seminvariants, then we try to find a fundamental

system of invariants.

two covariants of the same order with

If there be

the same leader, then their difference has the factor L, and we can reduce so that it is sufficient to consider only one covariant of a given order with a given leader. would refer the reader to the original papers

We

for further details*.

17.

Dickson's

Let /i in the Si

and

(a?i

.

,

.

.

,

Method

#m ),

,/e (#1 ,

independent variables let

qt

-

q/g {

.

Formal Covariants.

for

st #,) be forms of total degrees s^ xm Let g be the H.C.F. of q and Then for any non-zero element /o, of the . . .

.

.

xlt

. . .

.

,

...

,

,

.

* p.

Dickson, Trans. Amer. Math. Soc. vol. 14, 154 (1919).

p.

,

t

299 (1913) and Glenn,

ibid. vol. 20,

29

MODULAR INVARIANTS and hence

[/ (P^

Thus /? has a

definite value at every point

to is

. . .

,

,

p0?Xh

[ft (#1 ,

II

-

-

,

#,)?<

whose coordinates belong

We have seen in 4 that all such points are conjugate, that ]. they are permuted by transformations of the group; hence the values GJ?[p

n

(/*i

be permuted by transformations of the group. Now let will take different ,/*< then ,/*) be any function of /^

will

/?* also .

.

.

,

,

. . .

;

P points values
p must be a formal co variant, since the merely to permute the different points. As an example we consider the binary quadratic modulo 2

quantities fa any transformation .

,

.

.

<

eifect of

,

is

/=

2 rt

#i

+ <%#i#2 +

a<2 %/.

The only points with coordinates from the GF[2] are (1, 0), (0, 1), a a z a + a^ + a 2 Any symmetric (1, 1), for which/ takes the values invariant. We easily infer that a formal be must three of these function a l9 # 2 + a2 + a a + ^0^1 + %2 and a^a, (a + i + 2 ) are formal in(}

,

.

,

Other examples are given by Dickson*. can also find certain covariants by an extension of this method. Let y lt y 2 ..., y m be a set cogredient with x^ x, %m'> then

variants.

We

,

,

is

an invariant whi|h we can use as an auxiliary ground form. Now, invariant of x] be an w-ary ground form, then a simultaneous

if f(a,

x) and of /(, y)-

f(a

<

y

obtained by Dickson's method will be a formal covariant

A modification of these methods can we

see from

7 that

be used in simple cases. If p =

every function which 1

is

invariant under

= #1 + #2

and under

*

Tran. Amer. Math,

Soc. vol. 15, pp. 499

ff.

(1914).

2,

MODULAR INVARIANTS

30 is

a formal co variant.

Consider the case

p=

m = 2,

n=

l

of the

quadratic

/= a^x-f +

aiffii

^ f becomes / where

Under

a Q = a Qi III

i,

A function of the a's which is invariant under Ji is called a SEMINVARIANT. It is easy to see that

are seminvariants.

Any

function of these which

is

symmetrical with

respect to a and $ 2 will be invariant under T.3 also, and will therefore be a formal invariant. Thus we have the following formal invariants 01

18.

+

2

03

-

Symbolical Representation of Pseudo-isobaric Formal

Covariants. If we consider the system of formal covariants of the binary linear form modulo 3 given in 14, it seems at first sight as if every formal covariant could be expressed as a rational function of symbolical inner

and outer products. This however

Then

K= a^ + a%

is

known

is

not the case.

Let

to be a formal invariant of

/

if /;

=

2,

but

it

cannot be expressed as a rational function of symbolical inner and outer products. Miss Hazlett however noticed that if we write f ax ftxyx,

K*

then

The remainder

in

a.

psn

(/?

of this paragraph

what Miss Hazlett* did Let

2

S (a2 r)

- a (s)

.

y) is

+

(a/3)

(py) (ya).

an extension to the w-ary case of

for the binary case.

Then a^, am

.

.

.

,

a^

are simply a set of elements cogre-

C, a pseudo-isobaric formal covariant, oj, consist of terms C<. The weights of the different terms must be congruent modulo q. Let the weight then of (7< with respect to x be wl -f qWu,

dient with the set

where

w

l

Suppose variants.

is

...

,

.

Let

the weight of the term of least weight with respect to

first

that ^=(=0.

The weight

of

Now C

pn

xl

.

and therefore 3<7f are formal con is p (wl + qWu). Now

0?" with respect to x l

Trans. Amer. Math. Soc. vol. 14, pp. 300-304 (1913).

MODULAR INVARIANTS

qW

replace

lt

symbols of the sort af, /3f

corresponding symbols of the sort

...

8J

,

,

C?" with respect to Xi

is

a*

ft^\

*,

We make

symbols in the terms where Wi

> Wy

TWN pn Wi.

terms

If there still exists

...,

as before,

',

...

8^, x^\

^

,

by the

x^. Each

...,

q, so that the weight of

The weight

.

Wi^W

lk

Wy

C pn

of such terms

which

for

C%

f

a further replacement of

The weight

.

n

sider next the formal covariant 2,0 l

x^

now

do this for every term C^*.

now p 2n w

,

1

replacement lowers the weight of the term by

We

31 n

is

we con-

,

of the term

Cfis

n n W"n < Wi and is (p Wi + q Wit) p if Wu ^ w l We proceed n all w and if for then on making p n Wu replaceli

.

,

W

l

l)pn

}

we find by af ..., 0%** by af by a^, a[ 2n n that the weight of every term is p Wi. If wl for any Wu> then we must go farther and consider %C? ", and so on until we have ments of the types

^,

,

Wu^p

W i
Wn. Then we have

for all

L

to #i in the symbols

a,

similarly with respect to

a<

!

),

a<

,^,

2 ),

--, oc

that

C

is

a w, ft /8W

...,

m and

finally

isobaric with respect,

#. We

proceed

we have that

2(7?*" is

...,

completely isobaric in the corresponding symbols. We therefore have the important result that if C be a formal covariant then C pns for some s is

congruent

&l \

#w

...,

f(a,

to

i.e.

,

an to

isobaric

polynomial of

the symbols a, a^,

a simultaneous congruent covariant of

*),/(", x\

p '"

...,/(a

ar),/(a,

^O, -,/(,

...

^

= 0, we

first

multiply

C

a^,

/3,

O

......

If

,

(18-1).

algebraic invariant / whose then the proof holds for JO, so that C is

by some

greater than zero, and congruent to the quotient of two such simultaneous congruent covariants of the ground forms (181). is

weight

Thus

if it

symbolically

be true that every congruent covariant can be represented it will also be true that every formal covariant can be

(l (2) represented symbolically in terms of the secondary symbols a \ a x(8\ As we have seen in 10 however, this has been proved for the binary case only. .

.

.

,

19.

We

,

Classes.

must now consider the theory

variants.

of classes

Diokson found this the best method

invariants.

and

characteristic in-

for dealing

with residual

Now if the coefficients of the ground form / belong to

GF [pn

]

MODULAR INVARIANTS

32 there will be a finite

number

of possible

/

J

If

s.

two of these can be

transformed into each other by a transformation of the group, then they are said to belong to the same CLASS. If they cannot be transformed into each other by a transformation of the group, then they belong to different classes. All the/'s can thus be separated into classes such that

any two/'s in one

class are transformable while

classes are not transformable.

quadratic

where

a, b, c

We

shall use as

any two/'s of

different

an illustration the binary

f = ax* + bocy + cy'\ belong to the

are evidently

GF [2].

/ = 0, ()

/ = 4

The

different possible types of

/

3

tf

,

.(191)

while the possible substitutions of the group are

......

We

tabulate the results obtained by transforming

in the following

manner, where

e.g.

T f ==/i 4

s

(19-2)

every/ with every

:

(19-3).

T

MODULAR INVARIANTS

We

therefore separate

/

,

C

33

...,/7 into the following four classes:

containing /<

*\

6i containing .A/3/5,

Ca containing /2/4 ft containing

/

7

/

6

.

,

1

(19-4} I

j

We

can proceed in this manner in the general case even where we have more than one ground form. For example, Dickson gives as typical members of two m-ary linear forms the four classes %i,

dx*

ft;

#1,

ar 1

ft;

0,

^

ft;

o,

o.

ft;

Classes under a group may be further separated into sub-classes by a sub-group of the group. Considering, then, these classes we see that we may expect a residual invariant to take different values for different

Indeed this gives us another definition of

classes.

a,

residual invariant,

namely a residual invariant is a function of the coefficients of a ground form such that it has the same value for two/'s belonging to the same class, but may in certain cases take different values if the two /'s do not belong to the same class. Characteristic Invariants*.

20.

We

now come to the important functions called CHARACTERISTIC INVARIANTS. The characteristic invariant Ij of a class Cj is such that it

/

takes the value 1 for every for any belonging to CJ- but the value other /. Suppose that any particular residual covariant V takes the values i? ..., v k _i for the classes ft, ..., ft- 1} then obviously ,

The

A ft

of possible invariants of this sort is manifestly p nje set of invariants lt ..., Hi are said to CHARACTERISE the classes .

R

>

,

number

total

ft

if

they form a criterion for determining to which class a parThus if an invariant Jii takes a different value

ticular form belongs. for it

for

ft

each

class,

then

it

which

and

Cj

it

all classes

takes the same value

and *

it

Suppose however that with the exception of ft and Oj then it will not distinguish between

characterises the classes.

takes a different value for

;

does not therefore characterise the classes.

Amer. Jour, of Maths,

vol. 31, pp.

349-366 (1909).

If

however

MODULAR INVARIANTS

34

we have a second classes Ct

From

definition

R 2 which takes different values for the and R^ together characterise the classes.

invariant

R

and Q, then

:

obvious that the characteristic invariants

is

it

characterise the classes.

We

now

shall

consider a few of the properties

of the characteristic invariants.

THEOREM. A ny residual invariant can be expressed in one, and only way as a homogeneous linear function of the characteristic invariants

one,

with

coefficients

PROOF.

of the field.

If it were possible to express the invariant in

two ways, then

we should have ...

=

whence

Now

for

or vt

=u

is

t

(>

-

u ) 7 + (X - u^ /, +

a class C, we have .

proved.

7= f

and

l

+Wfc-l/A;-l

...

........... (20*2),

+ <X_i - %-i) /&-i

= () (j*i\ Ij

.(20'3).

so that

= 1^-1*1

We get this result for every value of i and thus the Jheoreru We also see from this theorem that the number of homois equal to the number notice however the non-homogeneous

geneously linear independent residual invariants of non-equivalent classes. relation

We

l=7 From the relations

+ 7 + ...+7 _ 1 J

fc

definition of characteristic invariants

.................. (20'4).

we obtain the

following

:

I? = It,

By means of

these,

any

7,7;

=

0>jO

.........

(20-5).

relation between the characteristic invariants

can be reduced to a linear one which can be made homogeneous by means of (20*4). Thus any relation between the 7's can be represented as

2c 7,--0 t

........................... (20'6),

whence every c t is zero. Thus (20*4) and (20'5) are the only relations between the characteristic invariants. Dickson* has given two rather artificial

case.

If

formulae for these characteristic invariants in the general coefficients CD ... c, we have

a system of forms has

,

/.-SD^l-^-cf)'} where the sum extends over *

Madison Colloquium Lectures,

p.

all sets

.................. (20-7),

of coefficients cf\

...,

cf\ of the

13 and Amer. Journ. ofMaths.vol 31, p. 339 (1909).

MODULAR INVARIANTS

35

various systems of the class Gk The second formula making use of Lagrange's interpolation formula

is

.

s

7 =

H

2

fc

cw... c

C

l >n

_

-^

J

obtained by

.................. (20-8).

j=ic) '-c,.

Both of these expressions give

/ =n(l-c

c

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

)

(20-9).

These results are not of much practical value, but they prove the existence of the characteristic invariants. 21.

Syzygies*.

Let Li

t

...,

L<j

be a

full

system of residual invariants, then every

L a Let must be a function of L\ / =
characteristic

. . .

.

,

,

t

II

>

||

?

expressible as a linear homogeneous function of the characteristic of invariants so that every syzygy can be represented as a function the characteristic invariants, which is equal to zero ; thus is

G

(/

,

...,

Ja-OllO.

But we have already shown that the only relations between the /'s are must be a result of combining (20'4) and (20'5). Therefore G and (20-4) (20'5). II

Now

suppose KO,

A-l

KH

S a /x and

II

x7l

A=0

...,

^

fundamental system.

Kit-i is another

Since

k-l II

S aM7 /M then ,

^=0

jff"^

II

i %;^M/A

A=0

H

*S r=0

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

(211),

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

(21'2),

C^JT,

A;-l

where

c vjh

= c vly= ^ aKh a^A^ v = A

if J. A>/

be the cofactor of a Kv in a divided by a \. of the jfiT's can be represented as a linear homo|

|

\

Thus every product

geneous expression in the JT's themselves.

If

now

a syzygy between the JTs then by means of (211) we can reduce the syzygy to a linear relation between the JT's, or else both sides vanish. Suppose then that this relation is expressed by is

5

...(21-3).

From

Prof. Weitzenbock's notes.

MODULAR INVARIANTS

36

K

But

v

k-l

2

II

we have

a\ v l\, and so

x=o k-l k-l

2 ftfl^A

2 v=.0

I!

ft.

\=0

Therefore

&Ao+&0;u+---+Afe-i0A,*-i so that

ft v

II

/?2Av

II

(*

II

ft(A* + A lv +

...

= 0,

1,

...,/;-!)

+-4*-!.,),

A

and substituting

in (21*3),

V

~2 v=Q

(21 '4) and and (20*5)

\J^ + ^+...+^*-i,,)#"JI

1

.........

(21-4).

(21*1) we call ground syzygies. These correspond to (20 "4) in the case of characteristic invariants.

THEOREM. If a residual invariant J takes the value J for the class d = 0, ..., # 1), and if g of these values J t are different when 2^g^p, %

(i

J satisfies a

then

congruence of the gth degree, but satisfies no congruence

of lower degree.

From

PROOF.

J JI +JI Q

\\

Q

1

1

+

...

+/A _ / _ 1

A;

1

and

(20'5)

we have

Give p the values 0, ..., k- 1, then since (/-/p )/p one Ip must be congruent to 1, one factor (J~Jp ) must be congruent

Jfp

II

to 0,

Jplp or

||

0.

and therefore

Let J^,

...

,

J

f

be the g different values of /, then obviously

is a congruence satisfied by J and is of degree g. If J satisfied a congruence of degree s < g then the congruence could not have g different t/p and so there is no congruence of degree less than g. roots J Pl y

. . .

,

,

22. Residual Covariants.

Every formal covariant

is

a residual covariant.

We

can of course

reduce the non-symbolical coefficients of a formal covariant by Format's Theorem, so that the function remains a residual covariant but is no longer a formal covariant. The residual covariant thus obtained turns out in many cases to be zero. The most obvious method then of finding residual covariants is first to find formal covariants by any method suitable

and then to reduce the non- symbolical

coefficients

by Format's

Theorem. In the case of residual invariants we can use the following theorem

MODULAR INVARIANTS

37

we have a full system or not. If the residual invariants Hi completely characterise the non-equivalent classes, then they

to find whether

R

l

^

...,

form a full system of residual invariants. Let

cl

. . .

,

,

c8

of the forms in the system S. For the resulting p the c's let 1A> RI take the distinct sets of values

sn

be the

coefficients

sets of values of

-

,

There are thus ^non-equivalent classes in the system S; and by hypoby the values JRi, ... Mm). If an invariant Q takes the value Q t for the class ft, then thesis the eth class is uniquely defined

,

R

Hence any invariant can be expressed as a polynomial in the l9 ... RI n Thus the invariants RI, ... RI form with coefficients from the GF\_p a full system.

The converse

of this theorem

is

,

,

~\.

also true, for if

system every characteristic invariant

R

l} ...,

RI are a

must be a function of

full

RI. Since the characteristic invariants characterise the classes, so also must J?i, ...,

MI, ..., RI. 15 and In

a

2

17 we found by different methods that a l and + a* + $0^2 + $o#i + #1^2 were formal invariants mod 2 of

Therefore using Fermat's Theorem

we

a + c + ab + bc + ca + b%y + cy* moreover

find that b,

are residual invariants of the modular form aar

they characterise the classes, for they take the following values for the different classes

:

b

ft

1 1

ft ft

1

1

Thus b and a + c + ab + bc + ca form a full system of invariants. As will be shown in a later paragraph, they also form a smallest full system. From the theorem of the next paragraph we have a means of obtaining a full system of residual invariants if we know a full system of formal invariants. Miss Sanderson's Theorem has not, so far, been

extended to covariants, and so at present there

is

no definite method of

proving whether a given system of residual covariants

is

a

full

one or

MODULAR INVARIANTS

38

full system of residual covariants system of formal covariants. Glenn* has given the following full system of formal covariants of /= a^x* + a^x^x* + az Xz, where p = 3

not.

It

seems likely however that a

can be obtained from a

full

:

A = a^ -

$ = #/ + xfxf + x?x + x -

ft

= (d^dl -

8 1

)

#1*

+

(#2

~

= (02 +
l

1

2

8 1

+

I

)

(1

8

+ x-^

<>

^0

02)

2

8

(^0

1

-

4

<^2

+ a*a - a? a* + a Q af) x^x^. + (a* a* - a^} x*

We now suppose that

>

<*i> <^2

are elements of (^^[3] ; this gives at once

*all/, J5

We

shall

ii

4ll/4,

4

ii

4

it

o.

however expect to find relations amongst the remaining C19 C,, 04,/4,/e, t, 4, thuS/6 7s -4 ft'.

/, A, J, T, L, Q,

II

Dickson t gives as a

full

/, A, ft, ftj,/4

Also

T

in

II

II

II

,

(a +

,

., Q, and

(o 8

02)

,

II

system of residual covariants

2

^i

q^(a Q + a2 )(a

+ <*22 + ^0^2)

2 2 (i + o2 ~ 1) + (o + a) Oo 2 z 2 q + 2a 4- flo^ + a a2 + a z a Q + a^a? +

(o + g+

(

a)

+

^2)

W

+ o2 -

1)

So we see that every residual co variant in this case can be obtained from * Trans. Amer.

Math. Soc.

vol. 20, p.

154 (1919).

f Ibid vol. 14, p. 310 (1913).

MODULAR INVARIANTS a formal co variant

39

the methods of proof that the above systems are full

;

systems are not sufficiently general to be included here.

23. Miss Sanderson's Theorem. Miss Sanderson's Theorem * can be stated as follows

:

To any residual

of a system offorms under a group F with coefficients from n the GF[p ], there corresponds a formal invariant I under r such that, invariant

i

for all sets of values in the field of the coefficients of the system offorms,

1

We require the Lemma ...

g lt

Let a lt

,gr given elements of

the

..>

L

1!

,

a r (r >

GF[p

n ]}

1) be arbitrary variables

and

gr *Q, then the determinant

a1

<* p(r-i)n *!

is divisible in the field

C

" ">

*

by the determinant ...

!

ar

.(23-2)

al

and

the quotient

Q-

N has -^

the properties if

Q=

if

!

!

= #,

...,

ar =

=

...,

ar =

.(23-3),

e l} ... er are elements of the field not proportional to glt ..., gr "We refer the reader to Miss Sanderson's paper for a proof of this lemma. We consider a system S of forms /i, ... ft inm variables with coIf S' efficients Oi, -..,a r We separate all such systems into classes Ct

where

.

,

9

.

.

a particular system of forms//, ...,// and if k is a constant, then we shall say that the system //, ..., /*' is a MULTIPLE of 8' and shall such that if s is in d then denote it by M', Now let c{ be a sub-set of is a multiple of s is in ciy and such that every system $ in no is

d

d

multiple

of some system P

v

=

1,

mc

then y must *

t

a primitive root of the n be an integral multiple of p - 1.

.

Letp be

Trans. Amer. Math. Soc. vol. 14,

p.

GF[p

n

Let

e{

490 (1913).

],

i.e.

if

be the

MODULAR INVARIANTS

40

e

smallest exponent for which s' and p is' are equivalent, where n1 and system in d. Then e t is a factor of p e

s',

p is

,

are all contained in ft. If S" e

p

*/S"'

in d, for

in ft,

"

pV

possible,

......... (23'4)

fe<

in ft is equal to p V times a system pn - l - JceiS'" is (0 ^ /<^). Since p

= p**f + V"

we may write S in ft. But ^ is the smallest exponent hence I = 0, and we can write f

s"'

any

is

ft. For S" Moreover any system 8'"

e

into p iS".

= />*-!)

is

a system in ft not in the set (23*4), then e e may be transformed into $', s' into p *V, p ts'

be in

will

(dt e t

pPi-Wta'

. . .

,

s'

is

for

which

this is

where 2 denotes an aggregate, and dp* 6* is the set of systems obtained by multiplying each system in c by p**i. If ^= 1, ft will contain all the t

multiples of the systems in

c,.

In general there are

e t different classes,

dt ke Cu= 2 dp +l i

(/

=

1,2,

0,

.

..,<-

......

1)

(23'5),

7f=l

formed on the sub-sets

d

1

c
respectively.

Thus a complete 1 ) and (700 ei

h I = 0, 1, given by Ca (i = 1, which contains only the identically zero system. We have seen in 20 that all invariants of the system list

of the classes

is

.

.

.

.

,

.

;

.

,

S of forms can

be expressed in terms of characteristic invariants %, which have the value 1 for the class CM and the value for every other class. To prove our

theorem

it will

reduce to it

be sufficient to construct formal invariants /# which

% if the coefficients are in the field.

For

this

purpose we find

convenient to employ the invariants

-,9rW

(t=l, ...,A)

...(23-6),

gi-ffr

where the summation extends over the different sets of

When

coefficients in

c*.

the variables are transformed, the coefficients Oi, ... ar undergo an induced transformation which is also linear with coefficients in

GF[p

9

n ~\.

Ji

is

a formal invariant, since the numerators are invariant

apart from a factor which is the c^th power of the determinant of the induced transformation and the denominators are permuted among themselves apart from the

same

factor.

Consider any particular denominator ... a r a, (23-7).

MODULAR INVARIANTS Since the

41

'

coefficients of a system of c^ this system After the transformation ai = ^c jAj this becomes

also in C&.

gr are the

,

is

f

(2 means 3

)

3=1 J

\

(23-8),

where the

may

G

19 ...

,

Gr are also the coefficients of a system in

be written p^gS,

a system of ct

Hence

.

fc

...

,

p '#r", where

"

,

gr are the

C& and hence coefficients of

(23'8) becomes

d ze occurs Since p < d '=l, this denominator apart from the factor |c e di is the factor in sum and the the denominators \Cg\ (23 '6), among cancelled by the same factor which is brought into the numerator by |

the transformation

We by

shall denote

Ji(Qfc).

:

"

hence Ji is a formal invariant. the value which Ji has for systems of the class

Then obviously by

Since Ji (C^)

=

0,

The theorem of

we may put

(23*3)

and

= p and l

Ji

(ft- )

easy to show that

it is

20 implies that any invariant

V can be written

V = ^Vjk Ijk where V ( (7^) = v^ ,

Cy

(23'6)

.

?fc

We can therefore

determine the /# from f'"'

/f= for the

For

Coo

determinant of the coefficients can be shown to be non-singular. we have

- 2 01

-0

MODULAR INVARIANTS

42

We now

have a theoretical

the formal invariants the class cjk

.

Jjk

if

not a very practical method of obtaining ijk of

which reduce to the residual invariants

The theorem

is

therefore proved.

Miss Sanderson's Theorem shows us that

we can

formal invariants

For

residual invariants.

have the

full

full

system of

a^

-

< + a/c^

a, a,/,

2

+ a/a/ 4-

a/,

2

and af + a^a^ + a2 respectively when the a's belong to so that a full system of residual invariants of the binary linear

which reduce to

form with coefficients from GJ?\3}

A

we have a

system of formal invariants c^'oa

GF[3],

if

easily obtain the corresponding full system of + a 2 # 2 mod 3 we the binary linear foYiaf

is

given by

residual co variant does not take a definite value for each class

we cannot

write a residual covariant

7 C + /!
(7=

and

so

C as .

.

,

/fc-iCfc-!.

Miss Sanderson's Theorem therefore cannot be directly adapted for covariants. It seems likely, however, that every residual covariant can be obtained from a formal covariant; and in 22 we have seen that this is

true in one case.

24.

A method

THEOREM. Let

of finding Characteristic Invariants.

K

Ju-\ be k absolute rational integral residual value of for the class CK where there Vi; these k invariants will form a fundamental ...,

,

invariants suck that a Xp are k classes

C

,

...,

system if and only if a x? \

By

K

is the

ft

-H-

0.

\

equation (201), JT,

n*^ a Al ,Jx A=l

(WO,

...,

A-l)

.........

(241).

If a Xp 0, then the JT's are not linearly independent and hence do not form a fundamental system. If |#A P |-H-0, then we can solve the equations (241) for the characII

|

|

teristic invariants,

and obtain 1

Jx

ll'i

^JT,

(A

= 0, ...,*-!),

i/=0

A

Kv is the cofactor of a Kv in |a Xl/ |, divided by \a^\. It follows that the JT's are linearly independent and form a fundamental system.

where

MODULAR INVARIANTS From

43

theorem we have an easy method of finding the characterprovided that we know already some other fundamental

this

istic invariants,

system. We continue with the example of the binary quadratic modulo 2. Dickson* has shown that a fundamental system of residual invariants for this case is given

by

"We can tabulate the value of each

K

for

each

C as

follows

:

ft 1 1 1

the reciprocal determinant

is

And we have

I = K% = abc. B

25. Smallest Pull Systems. a fundamental system cannot be a smallest full can be of the relation (20'4). Indeed any invariant account on system It is obvious that

K

expressed as follows

:

m 1 m 1 m 1 \ / JT= 2 %/,= 2 ijIj + iJl- 5 /,) y=o

=

o

+ ft

\

j=i

fc'o)

Ii

+

.

/

j=i

+ ft-i ~

Madison Colloquium Lectures,

o)

Im-i

p. 29.

-

MODULAR INVARIANTS

44 If

we

are considering the field

GF[p

then any invariant can take at most invariant can characterise at most

can characterise at most

k classes system

will nt

be

t

n ( t+1

k ^p ). k\ i.e. p a full system with only in a smallest full system

n

different values, that

is,

any

It follows that s invariants

different classes.

Suppose that there are

of invariants in a smallest full

We

the greatest power ofp n which is less than shall now show that there always exists

+

elements ; hence the number of elements

+ 1, where

<

ns

p

p

whose elements are

~\

classes.

minimum number

then the

;

p

n

n

t

t is

is

1

t+l.

The method employed in finding the elements of a smallest full system can best be illustrated by studying a simple example. Let us suppose that there are 12 classes ft, ft, ..., fti, and that their characteristic invariants / 7j, ..., /u are known. We shall also suppose that the field is GF[S]; the elements of this field are 0, 1, 2. Now ,

3

2

<11 <3 8 hence t = 2, and ,

the smallest full system. the different classes.

there will be three elements

Let

A, A>

A

A> A> A m

take the following values for

/"Y/>/'>/~y/'Y/Y/"y/"Y/'>/"Y/'y/"Y

Since the /'s are

known we can

find the D's at once.

that the D's form a smallest full system. that equations (25*1) are soluble for the

It is easy to see

At first

sight it does not seem /'s, but this is not difficult in

view of the relations (20'4) and (20'5) by considering expressions such 2 2 = = /io-/n, hence , /io + -/n so A, -.. E.g.,

as

A A AA + (A A)

that

AA

2

As an example we

consider the binary quadratic

be two invariants in a smallest classes.

Give

(AA)

- - /io-

A and A A A

system, since from the following values : full

G0

t/i

C/2

t/g

o o

i

o

i

o

i

i

mod

2.

There

will

19 there are four

MODULAB INVARIANTS then, from

45

24,

D

= J1 + l3 = b(l + ac) + abc = b, Dt = I2 + It =l + b + (l + a)(l + fy(l + 1

c)

+

giving the same full system as that obtained in

22.

The method

perfectly general

illustrated in the above

examples

is

Further

26. Residual Invariants of Linear Forms.

The

following treatment of linear forms

~ an #! + a li

#2

+

...

due to Dickson*. Let

+ a im x m

4=n(l-a? r) r

andput There are only two classes

C

form and

l

for

a single form

containing every other form. ,

also

ia

is

/ = ^LI,

I

1

O

..................... (261).

C

119

containing the null

We have readily

=1-A

I

;

seen to form a smallest full system. must now consider the classes of a pair of forms 4 and 4-

AI

is

We have and in this case we choose 0, 4 ^ 4 = 4 = 0, Ci has 4 If 4 and 4 are linearly independent, then we have the as a type 0, ^ class C2 and choose as our type x^ dx^ where d= 1 if w > 2 and e = Z>12 the discriminant of the forms if m = 2. We have also the classes C8 C where 4 is c^ and as type we take x^ cx^. For the case where m~2 In general let k be a instead of one class Ca we have q classes 2 (d), n non-zero element of the GF[p ] and write l^li Mj. Using kpn ^ we have We

the class

if

(7

.

,

(

)

-

II

A V = n [i - (. - fa^yg

ii

r

and the write If

coefficients of

powers of

^ + ^ (^ Jc

must be

1) **

+ s i=l

invariants.

8* In particular

F0 = $0i.

w > 2, the values

are as follows

of the different invariants for the different classes

:

Class

C

3 (C)

*

Proc. Lond. Math. Soc.

(2), vol. 7, p.

430 (1909).

MODULAR INVARIANTS

46

From

this table

we have

/ = J.i^2, /*) =

For the case

readily

/i=-4i

-A

(1

l-(c-F ) (c*0), w = 2 the value of /

= A<2 (l~Ai), 7^1-Jo-/!- 5 /8(c)

2 },

9

12

Iz(Q)

.

o/z>

is

2
l-(d-DK}.

A

smallest full

A,,A

is

system

,V

2

12

,

given by

A A, Vu D

w>2;

if

lt

,

12

w-2.

if

,

Smallest fall systems can easily be obtained from the above characDickson gives the following fundamental systems of residual invariants of a pair of linear forms 4 and 4

teristic invariants.

:

ItA^A^A^s, F A l} A^A,A,, FM *,Aa*

*

(=1,2, = 1,2, (*

ia

We find the classes of a system We say that JEJpar follows E rst

...

First, let

s~cr, #
...

of if

not every

w>2, w-2.

0)

...,

...,g)

X binary linear forms as follows r

:

D

u

vanish.

(r

<<*),

Let

D

rs

be the

first

which does not vanish, then

^=

A*=0

(i
J9r.*0,

and by an obvious transformation lr =-

Since Ar = As =

D

Since

l{

=

rfc

=

l

<

-cx2

Is

,

c

,

r),

^

is free

of #1

(r
is

free of

#2

Ir

0,

x

(^

4 - CkMi

x-i)

,

4= }

Secondly,

Lastly

By

Now

we have the

giving

possible,

every

c,

the

we can

D$

and ^ 2

.

.

therefore

;

(i
= Drs

lt

ca?2 >

vanish but not every

the

<#/s,

in the above

the

e t '8

manner

=

/,

as type

d

t

......

C4=0)

(26'2).

then we can take as

class C$ containing only null

CAT'S,

we can take

forms

and the m/s,

......

all

(26 '4).

the values

find all the different classes.

the products

......

(26-5)

MODULAR INVARIANTS

47

are linearly independent in the field; also the number of these invariants c is equal to the number of classes r f'. Similarly

A

\

H

l-Af)

Fft

(/*,

= 0,

1,

...

,

q

:

not every

= /*,

0)

are linearly independent and of the same number as the classes of type The products (26 "5) associate the invariant 1 with the class C

B. We

.

vanish for the classes of 13 and

A

also for those classes of

A

which

B

;

.

given by I,

^

m

itself.

(26-6),

(26'5).

method Dickson gives a fundamental system of linear forms, but for this we would refer the reader to the paper He also treats the case when A, < m.

By an extension of X

,

(26*6) vanish for the classes (7 and those of B which follow f fundamental system of residual invariants of A. binary linear forms

follow-4 rs

is

(7

this

27. Residual Invariants of Quadratic Forms.

Our

task in the finding of residual invariants of quadratic forms

first

We

see at once to separate the general quadratic form into classes. that this is a much more difficult task than it was in the binary case. 2 and p > 2. shall find an essential difference between the cases p

is

We We shall follow First let

p>

Dickson 's methods*.

2,

then we can write the general quadratic form as

m

and we can choose b u 4= with perfect generality. Let the determinant B = & y be of modular rank r, i.e. every minor of order exceeding r is congruent to zero but not every minor of order r is congruent to zero. .

|

Now

|

let

bn^i

III

bu then under a transformation of matrix t

1

.(27'2) 1-

we have q m = b u x? + <, where is independent of x way we can ultimately replace qm by <

0,

* p.

Madison Colloquium

263 (1908).

Oa*0,

.

...,

Proceeding in this

a r =f=0)

...(27'3).

Lectures, pp. 4-12 and Amer. Journ. of Maths,

vol. 30,

MODULAR INVARIANTS

48

Now every a thus obtained can be written congruent to a power of some primitive root p of p. After applying a transformation of deter2 minant unity which permutes the #Y2 as #y we can assume that a^ are even powers of p while a s+1 ... ar are odd powers of p. The trans. . .

,

,

,

. . .

,

,

,

formation

(27-4)

is

of determinant unity.

If r

< m, transformations of the above type

replace 2

i#i +...+0,.tf

This in turn

may

2 .

/

by xf +

...

2

2

+%i + p(% s+1 +

...+#v )...(27-5).

be replaced by one of the forms X\

+

...

ri

r _l

\

f,,.

(0
/,,._

n

x

............ ^ (27'6)}

by transformations of the types

^^

\

l

;'

where a u and a^ are chosen so that (*i? If

r~m,

+

3

in

a,/)-

p.

2 then transformations of type (2 7 '4) replace S a^i by i 2 ,r/+ ...+#s + p(#Vn+

+ ^2OT -i) + o-^v

5

...

......

(27'8),

not necessarily equal to p. If there be an even number of terms with the factory, transformations of the type (27 '7) reduce (27 '8) to

where

o-

is

z 2 xf + x2 + ...+x\,- + Bx m

..

1

............. (27"9)

an odd number, to

or, if

^12 +...+^ 2m _ 2 + P^m-l + p-

1

^m

2

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

(2710).

(27*10) can be transformed into (2 7 '9) by the substitution ~~

^m-i

p

ffirn. 5

#w = P~ ^m-i> l

x = Xi t

if

Or

B

(i

3=m,

i=^m-

+ III

p

\ " #m -i = a m -i, m-i^m-i - a^jn-jp 21 + %m &m,m~im-l &m~l,m-iPm, ~ 1),

1

by

^,

1),

49

MODtJLAB INVARIANTS where 0U-i,m-i an ^

flm,m-i are /rj2

in the case

where I?

chosen such that ^>-l + 2 '2a2 HI i_

(,ttm-i,m-i

P

.,,

**m,m-i;

n

P

l

in

p~

.

Thus every quadratic form q m modp (p >

2) can be reduced to one of

the forms of (27*6) or (2 7 *9) by transformations with determinant unity. For the proof that none of these forms are equivalent we refer the reader

Madison Colloquium Lectures, page 8. cannot of course hope to classify every possible modular form of the m-ary quadratic as we did in the example of 19. In the more comto Dicksou,

We

plicated cases

we must be content

to find a specimen of each class,

and

we can by

find whether or not a system of invariants characterise the classes examining the values which they take for the specimens of each

class.

The

B and r do not distinguish between the two classes and they therefore do not characterise the classes. known * that a symmetrical determinant of rank r (r > 0) has a

invariants

of (27 '6) It is

non- vanishing principal minor of order we can take this minor as

M=

bn

...

r.

b lr

By

a suitable transformation

-BfO.

bn

n

a function of r variables, arid proceeding by as before with only r variables we can reduce qr to It is possible to replace qm

Now,

let

M be congruent

qr

to p

zl

or

a+1 /o

,

then the transformation

M

2 = a in the gives us the two forms of (27*6) and pMa-i) or p + cases. But these are 1 to and 1 respective congruent respectively. q^l 2 distinguishes therefore between the two forms of (27*6) and we

M

shall call the non-equivalent classes

Cr>l and

Cv

t

_i respectively.

Let us now consider the functions

*

Dickson, Annals of Maths. Ser.

2, vol. 15, pp.

27-28 (1913-14).

MODULAR INVARIANTS

50 where

M

M

k denote the principal minors of order r taken in any lt ..., and the N's are the principal minors of order greater than r. is easily seen that A r has the value + 1 for any form belonging to

order, It

the class

Cril

-

the value

and the value

for

-

any form belonging to the

1 for

any form

contained in

riot

Crtl

or

Crt -

class (?r,-i; 1

:

thus the

following invariants characterise the classes

is

m=

"

J2)

^3.1,

as

A m -i^lJ

5

seen from the following table which illustrates the case n 3: Class AI Specimen A% B

C

^

Ci,-j

2^ 2

6\3

tf^+tfa

02.-1

^r+i^y

V*=i

aY'

63-2

x? + xf+2j'f

Jtl

2

mod

p - 3,

1

-

full

1

0*0

2

1

0-1

+ av'+tfa 2

since all the transformations

minant unity, we have as a quadratic

1,

000

ft

Thus

=

1

002

employed have been those with detersystem of invariants of the ternary

3,

A A lt

t

,

B.

We notice however that certain of the classes (7#, are equivalent under transformations whose determinant is not congruent to unity, so that a full

system of relative invariants A

-/l,

and the non-equivalent ' y o>

is

A

./l 2)

given by A -^mt >

classes are

^1,15 VI, -l

''2,l>

^2,-ij

?

^m.U

^w,-!'

of the quadratic where p = 2 is essentially different 2. For the separation of the classes we refer from the case where

The treatment

p>

the reader to Dicksoii's papers. He shows that an w-ary quadratic either reducible to a quadratic in less than variables, or else it

m

reducible to

if

m be odd,

or to one of the two forms

#!#2 + #3 #4 +

^^ provided that if be even.

2 4-

8

+

. . .

-f a:

2

84

S

+

...

+ #OT -

m -.iaym +

+

. . .

+

c

2W ~ l

8

H 1,

m

For small values of

m we

can of course proceed as in

19.

is is

MODULAR INVARIANTS

51

Cubic and Higher Forms. The quadratic case is rather difficult when compared with the linear and we may expect that the separation into classes of the cubic, quartic and higher forms will present great difficulty. This can only be accomplished without enormous labour in a few simple cases. Thus while we may find by different methods a number of invariants, it is not possible in general to say whether a system is full or not. The separation into classes is really a subject in itself and we would refer the reader who is interested to a list of papers on the subject which Dickson 28.

;

gives in the last chapter of his History of the Theory of Numbers.

29. Relative

unimportance of Residual Covariants.

Nothing has been said so far about finding full systems of residual co variants. Indeed there is no method known which can be applied to a general case. Neither of the two methods which can be used for invariants hold when applied to covariants. We notice however that classes of modular forms are completely characterised by rational integral invariants and therefore an invariantive property of a system of modular forms can be expressed by the vanishing of an invariant. This is in contrast to the algebraic case where a property may be expressed by the vanishing of a covariant. The difficulty of finding a means of forming a full system of residual covariants may be linked up with

the fact that residual invariants in themselves completely characterise

the classes of systems of forms.

30. Non-formal Residual Covariants.

The

subject of non-formal residual covariants has not yet been studied. n a's are arbitrary but the as are elements of the GF[p ].

In this type the It

is

seen therefore that these bear the same relation to the residual

covariants as the congruent covariants bear to the formal covariants. This gives us a clue to the study of non-formal concomitants. They

are obtained from congruent concomitants by reducing the coefficients of the forms by Fermat's Theorem. If it be true that every congruent

covariant is congruent to an algebraic covariant, then it is also true that every non-formal residual covariant can be obtained from an algebraic and in Also every noncovariant by reductions of the two types formal residual covariant is a residual covariant, just as every congruent II

covariant

is

a formal covariant.

It is seen

.

then that these non-formal

covariants are not of any special interest, but are merely those residual covariants which are obtained from congruent covariants

through reductions by Fermat's Theorem. 4-2

PART

II

Rings and Fields.

31.

A

such that for every pair RING is a collection of elements a, b, c, of elements #, b, a sum a + b and a product a x b are defined, where a + b and a x b both belong to the ring, and such that the following laws are .

.

.

,

satisfied

:

I.

a+

(i)

(b

+

e)

=

(a + b) +

=

b

(ii)

a+

(iii)

a + x = b,

b

+

c,

a,

has always a solution for x in the ring.

a

II.

ab x

c.

a(b + c) - ab + ac.

III.

We

x fo

shall only consider

COMMUTATIVE

rings where the following law also

holds.

ab = ba.

IV.

A is

ring together with the laws which the elements of the ring obey an ALGEBRA. An algebra which has no divisors of zero is called

called

a DIVISION algebra. That

then must b

is,

in a division algebra if ab

=

and

if

a 4=

0,

= 0.

A

FIELD is a collection of the elements of a division algebra for which every equation ax-b has one and only one solution for x provided that a is not zero. We shall denote this unique solution by x b/a. field is said to be FINITE if it contains only a finite number of

A

Every division ring JR, (i.e. a ring with no divisors of zero) is its QUOTIENT FIELD. The elements of this quotient field L are determined by all the elements given by solutions of the following elements.

contained in

type of equation

:

ax

where a =}= 0,

b,

provided that the further conditions hold a/b

(i)

= c/d

= ra/a r

(ii)

if

(iii)

(a/b) x (c/d}

- ac/bd,

(iv)

(a/b} +

= (ad +

(c/d)

:

if

ad =

be,

r belongs to the division ring,

bc)/bd.

MODULAR INVARIANTS

53

Every element of R is an element of L and we say that R is contained in Z. Every field which contains an element other than the zero element contains the unit element e which is defined by ex~x where x is any element in the field. There can only be one unit element, for, if ^ and e2

were two solutions, then would ei

a = a, where a 4=

Hence e^ab = e^ab and (^ If every element of a ring

We

said to contain L, " in ^ K] thus,

is

L is

0.

also

b

= b, where

b

=1=

It follows that e l

0.

= e%.

an element of the ring K, then

shall use the sign

^

K

to denote "is contained

K called a SUPER-RING of L and L called a M be a super-ring of L and at the same time a subM said to be a MEDIAL-RING between K and L.

L

;

2)

e>2

;

ab -

SUB-RING of K. If ring of K, then

is

is

is

Replacing the word "ring" by the word "field" in the foregoing definitions, SUPER-FIELD, SUB-FIELD and MEDIAL-FIELD are defined. If L be a

L^K.

K, we write

sub-ring of

K which are

If

L < K, L

=j=

K,

if

i.e.

there exists

elements of L, then L is called an ACTUAL sub-ring of K. The term "actual" is used in other cases with a similar meaning e.g. if L be an actual super-field of J5T, then L > JT, L^F K.

elements of

riot

;

Consider

now a

ring

R

containing the elements Oi, a 2 ..., and let x - 21a #* is called an /^-POLYNOMIAL. rt ,

be an arbitrary quantity, then

A

r

i

sum and product

of two /^-polynomials are defined in the usual then manner, obviously these ^-polynomials form a ring. This ring is and is denoted by called a POLYNOMIAL RING of [#]. By an obvious If the

R

R

extension, polynomial rings containing ^-polynomials in several variables can be obtained. Thus

R [#i, #2] = RI [#2], where Since the field

elements e + e = e,

20,

30,

-

.

K contains the unit element

2#, 30,

.

,

7^ =

ne,

. .

.

,

...

ne.

There are two

R [#J.

e,

it will

also contain the

possibilities here ; either

are all different, in which case the field

K

is

said

have the CHARACTERISTIC zero; or else they are not all different. In this case let me = ne, hence (m n)e = 0. Let p be the smallest value n for which this is true; then the elements 0, e, 2e, ... (p~ 1) e of m

to

are all distinct

and

K

JTis of characteristic

The

is

We

show that said to be of characteristic p. (p) the as case or may be. p by writing

or

K^

following theorems are easily obtained*

K

:

THEOREM 1. It is necessary and sufficient for the vanishing of the derivative of a rational function of #, either that the function is a con*

Steinitz, Algebraische Theorie der Korper, p. 46.

MODULAR INVARIANTS

54

stant provided that the characteristic of field of the coefficients be zero, or that the function be also a rational function of y = # p provided that ,

the characteristic be p.

THEOEEM

2. If f(%\ a rational function of #, has an r-fold linear (%) contains this factor at least (r^l), then the derivative the the to which times. If coefficients of f(x) belong has the (r- 1) field characteristic p and if r is divisible by p, then /' (x) has the factw at

f

factor

least r times, otherwise it

has the factor exactly r

1 times.

a polynomial in x pf and the cliaracteristic of f f(x) p -th power of a polynomial in % and (p \ are in a super-field L (p) of vice versa where the coefficients of

THEOREM

3.

If f(x)

the field be p, then

be

is the

K

<

The

proofs of these theorems are left to the reader.

32. Expansions.

K

A super-field 2 of a field KI can be obtained in the following manner. Choose an arbitrary element a which does not belong to jff^ and let the elements of } be &, /3 2 A- If a is to be a member of 2 so must ,

K

K

,

,

&

,

+ &, a + /3 2 2a + &, <*&, a 2 a3 + /? 2 a//?j ftn /a etc. In this way we obtain a number of new elements and each one can be expressed as 'a quotient of two polynomials in a with coefficients belonging to K^ The totality of such quotients, including the cases where one or both of the polynomials do not contain a, form a super-field J5T2 and we write JT2 = KI (a). A super-field of 2 is also a super-field of K^ and we write a n ) is a super-field of JTj JT3 = z (y) = K^ (a, y). Similarly K* (c^ a 2 Every super-field can be built up in this manner. As an example we notice that the field of all complex numbers is obtained from the field of <*

,

,

,

,

9

.

K

K

,

,

.

.

.

.

,

numbers by the adjoining of a single element i ~ v- 1 The above process leads us to use the word EXPANSION as an alternative for superfield. Some writers use the term EXTENSION as a further alternative. all real

.

An expansion is said to be SIMPLE if it is obtained by the adjoining It is easily shown of a single element which does not belong to that every element of the expansion K(x^ ...,#m ) can be expressed as the quotient of two elements belonging to the polynomial ring

K.

JET^i,

...,

#m l Hence

K
A

=

$#{#* and i

but that

A^Bj

B = S6

t #*!

We

Consider two

shall suppose that

K-

A-E

i

i.e.

for

some values of

,

a t *b %

.

Since

A=B,

MODULAR INVARIANTS

55

A - B ~ 0, where A - B is a polynomial vanish.

If

such a polynomial

expansion of

K

and x is said to be an algebraic element with respect algebraic element satisfies an equation with coefficients y

K. Every

to

from the

Let

in x which does not identically we say that K(x) is an ALGEBRAIC

exist,

field.

(V)

-

be the equation of lowest degree which

is

satisfied

by

(c

*0, n ^

1).

or

#?,

Now

4>(ai)

must be

(x)

=

#w +

1

+

Ciff*-

...

+

cn

=

irreducible in the field, for otherwise divisors of

would exist. Any element of K(x) can now be expressed as a polynomial in x of degree less than or equal to n 1 Indeed any element of K(x) is of the form f(x)jg(x) where /(#) and g(x) are JT-poly-

zero

.

be a multiple of nomials and where (/, g) = e, the unit element. If = since <#> (a?) = 0. If I be not a multiple of (a?), then, (a?), then <

<

g (x) may not be a multiple of polynomial h (x) such that h (x)
of

<

We

<

Hence h (x)

(x).

g(ai)

=

l

since

always possible to find a to equal unity plus a multiple it is

(a;),

is

<#>

(,r)

=

and

can now reduce the degree of/(.r) h (x) to n - 1 or lower by means 0. We shall say that x is of DEGREE n with respect

of the equation < (x) if to the field (,#)

K

<

is

of the wth degree in

^r.

For any w elements of K(x) we have therefore w equations of the type

+

.

so that on eliminating the x,

minant and there from

./T

^_ 1 r

2

,#

. . .

,

,

^;

ri

"" 1

n~1 ,

a

=

0,

we have a vanishing

deter-

therefore always a linear relation with coefficients between any n elements of K(x). have two such relations is

We

any n + l elements and therefore a homogeneous linear relation is obtained by eliminating the constant terms. Every element of an algebraic is algebraic with respect to K. This gives us another expansion of

for

K

definition of an algebraic expansion, namely, an expansion

said to be algebraic if every element of

L

is

L of K is

algebraic with respect to

K

is the field K. In some cases however every (x) is reducible, e.g. if of all complex numbers. In such a case no algebraic expansion is possible. Suppose now, on the other hand, that there exist no two elements A and B which are equal unless each a, is identical with the corre<

sponding

bi.

In such a case there will be no polynomial in

x which

is

MODULAR INVARIANTS

56

TRANSCENDENTAL expansion of K\ K. have therefore the following criterion: K(x) is an algebraic or a

equal to zero; and further,

We

x

K(x)

is

called a

said to be a transcendental element with respect to

is

K

transcendental expansion of according as e, dependent or independent with coefficients from K.

...

%*,

,

are linearly

An infinite

system

said to be linearly dependent or independent with respect to a field according as there exists or does not exist a finite of which is linearly dependent with respect these elements sub-system

of elements

is

K

toTT.

THEOREM

In a transcendental expansion K(x) every element not a transcendental element with respect to K.

1 *.

K

contained in

is

be an element of

If

where /(V) and g (V)

K(z\ then =/ (#*)/# (#),

belong to J5f [#*], and without loss of generality we can put (/, g) ethe unit element. Now if were algebraic, then there would exist an 9

equation of the sort

+

C

where each

c8

. .

belongs to K.

throughout by #

r ,

.

4-

r

Cr

=

(c r

* 0,

C

=t=

0),

and multiplying

Substituting for

we would have r

cff

- cQ g

or

+

CT

+ c fg r

-l

+ ...+crf r =

i

r

r- 1

=f[c^g

+

+

...

fr

cr

Q,

~l

l

thus g r would be divisible by f, which is in contradiction to (/, g) = e. Hence either is transcendental or else /and g are constants, in which case

is

contained in K.

THEOREM

2

1.

If n elements

so that the expansions

obtained,

(jb=l field

t

and n\

>

which

We

is

This proves the theorem.

K

Q (sr-^)

if every element

x\

,

#2

>

K K ,

x

1:

?'

.

,

. .

,

K

n-l

two elements of the corresponding polynomial ring. transcendental with respect to Q (x^ ..., ^-i, ^+i> will exist a relation

K

where each

d

is

*

t

Caa

+

-+e m a

=

,

K,

#)> then there

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

Steinitz, Algebraische Theorie der Korper, p. 23.

26.

to

a quotient of If now rk be not is

a quotient of two elements of

loc. cit., p.

respect to the

remaining elements

have seen that every element of an expansion

+ c^* +

K

x k transcendental with

obtained by adjoining the

e

K

^ ar adjoined to a field Q K^ n -i (n) n are

transcendental with respect to K^-i

is

tlwn is every element

?

(x^)

(32-1),

MODULAR INVARIANTS Hence multiplying throughout by the

L.C.M. of the

denominators we

being an element of K^ [x^ ... xk , ly can therefore rearrange the equation in powers of

can consider each

We

57

c t as

,

%k+l ... a?n ]. xn and thus ,

,

obtain

<%+ Crf +

K

C

...

+

6\< =

(32-2),

K

is from an -i] and therefore from Q (x xn -i) [a\ not transcendental with respect to j5ro (#], ... an -i). This is contrary to hypothesis, arid so xk must be transcendental with respect to KQ fa # does not occur in equation (32 *2), xk -i tffc+i #n)-

where each so that

xn

l

. . .

,

. . .

,

,

is

]_ 1

. . .

,

,

,

,

,

,

^

then we rearrange (321) in powers of the x with the greatest suffix and we proceed in the same manner as above.

33. Isomorphism. be a (I - 1) correspondence between the elements of two and if the products and sums of correspond to the and are said to be corresponding products and sums of R, then If there

rings

R and E

R

R

R

ISOMORPHIC. The correspondence is written R=-H. If b and a, b be R of from R and a elements corresponding pairs respectively, then a ,

b~T>, (a + 6)

(a

+ b)

and ab =

ab.

Isomorphic

fields are defined in

the

same way. Let L and

L be two expansions of the field K; further, let L and L in L correspond to the elements of be isomorphic if the elements of in L, then L and L are said to be EQUIVALENT. It is obvious that two

K

;

K

K

are equivalent, for /(#) /(/), simple transcendental expansions of hence K(x)~ K(y) and the elements of JTare self-corresponding.

34. Finite Expansions.

Let L be an expansion of K, and suppose that there exist n elements L which are linearly independent with respect to K, but that there do not exist n + 1 elements in L which are linearly independent, then

in

L is said to be of DEGREE

n with respect to K. This

\L\K\ = n\ [K:K] = The expansion The

is

said to be FINITE if

following theorems are proved

THEOREM

1.

Every

n

written as follows

:

l.

is finite.

by Steinitz*

finite expansion

is

L

of a

:

field

K

is

an algebraic

expansion.

Let [L

:

K] = n *

and

let

a be an element of

L

:

then, since

Steinitz, Algebraische Theorie der Korper, p. 34.

any n +

I

MODULAR INVARIANTS

58 elements of

between

0,

L

a,

are linearly dependent, there a n or

a2

,

must be a

linear relation

. . .

,

L

L

is an Thus every element of is algebraic with respect to K\ and algebraic expansion. The converse of this theorem is not true. If there exist n elements of which are linearly independent with

L

respect to K, then, since any n + 1 elements are linearly related, every element of can be expressed as a linear function of these n with coefficients from K. The n linearly independent elements are said to

L

form a BASIS of L.

Any

other n elements

terms of the basis elements

If the determinant

^

. .

.

,

,

&,

a n thus

..., ft n

can be written in

:

a ik does not vanish in the field, then ft and form a second basis of L.

.

.

,

.

,

ft also t

are linearly independent

2. Let K
THEOREM

that

M

The condition

M are

M

K

be

is

[Af:K] = v then

for, let

necessary;

)

v

+

1

elements

dependent with respect to K, that is, with coefficients from and therefore from L. is therefore a finite expansion of L. and are linearly Further v+1 elements of L are v+1 elements of of

K

linearly

M

M

L

K. The condition is also sufficient for, let \L K] ~ m and [M L] = n, where m and n are both finite, and let a lt ... a m be a basis of L with with respect to L. The mn ft be a basis of respect to K, and ft dependent with respect to K, so that

:

;

.

.

M

.

,

,

a finite expansion of

is

:

t

elements a,^ are linearly independent with respect to K, for otherwise would 3 Ci k a,il3k ^Q where each c* k belongs to K. In this case would ft

%

Yfcft

=

where y = 2 fc

c-^a

?:

,

but this

is

i=l

k=l

zero since the y's are from L.

If

impossible unless each y be

y k be zero, then

m 2 c^^ =

0,

which

is

impossible unless every c is zero. It is easy to see that every element can be expressed as a homogeneous linear function of the of 's, so that the a^'s form a finite basis of M. It follows that Mis finite ijc

M

a^

and that

THEOREM is

a

.._.

^

r

mn=[M:K] = 3.

If K(x)

finite expansion of

is

fc

v.

a simple algebraic expansion of K, then K(x}

K of degree

d,

where d

is the

degree of x with

59

MODULAR, INVARIANTS K. If x is of degree d with respect to K(x) can be expressed as follows ~ a ~ a + a x + ... + ad -i%d

respect to

a of

JT,

then every element

:

l

(at

}

Hence

e,

#,

...

,

x

d~l

K by

L

K

L of

4. An expansion can be obtained from

THEOREM if,

from K).

K(x) and hence \K(ai)

form a basis of

is

a

finite expansion

a

the adjunction of

\

K] = d. and

if,

finite

only

number of

algebraic elements.

This

is sufficient

K^K^fa Hence jk

K= K

for if

;

and

Q

JTa = *iC;V),

L=K

Q

K* = K^(j&

...,

K

(j1 ,j 2 ...,

,

. . .

,

/), then put

L^K ^K _,(j v

K

v

v ).

k is a simple algebraic k ^: and algebraic with respect to and is therefore a finite expansion of k ^ lt by k expansion of Theorem 3. Hence from Theorem 2, also is a finite expansion of

is

K^

K

L

-ZT

= K. The condition

also necessary

is

;

for let

L

be a

finite

expansion of

= K~K and [L then ^ belongs to L but not to K
(}

Q

:

jfiTJ

Q

=

If

n.

J5T

let

Q

is

,

2

Hence A^ < JTX (j 2 ) ^ L and so on. From Theorem 2 the JT ], [JT2 K^ are ascending integers less than ], \K$ or equal to n and hence for a certain finite v we have the condition

-^o

(^'i)

^TI-

degrees [JTj

:

^

:

'

K = L = K,(j v

THEOREM

5.

with respect

to

Let

K

JS

be

then

;

a

l

,j.2 , ...,A).

of elements a which are all algebraic also algebraic with respect to K.

collection

K(S}

is

8 is

a finite collection, then, from Theorem 4, K(S) is a finite and therefore an algebraic expansion of K. If $ is infinite and /3 is an If

element from

L = K ($), then

of elements a ls

contained in

K(a

l

,

K (S)

L

Therefore

av

...,

...,

is

/3 is

S

from

a rational function of a finite number

with coefficients from K.

Thus

ft

is

thus algebraic with respect to K. with from definition. respect to algebraic a,)

and

is

K

M

K

and let L be algebraic with respect to THEOREM 6. 'Let K^L^ be Algebraic with respect to L; then Mis algebraic with respect and let to

K

M

also.

For,

an element of an equation

if ft is

satisfies

a

Now

every at

-

fore of

a

ft

+

n

...

+ a n ft =

(a f

to Z, then

ft

from Z).

algebraic with respect to K, therefore from Theorem 4 a ) is a finite expansion of A" and therefore ft is algebraic

is

L' = -^r(a with respect to L'. j

4-

M of degree n with respect

Thus L' (ft) is a finite expansion of L' and thereK. Therefore from Theorem 1, ft is algebraic with respect to K.

MODULAR INVARIANTS

60

35. Transcendental

We

and Algebraic Expansions

now make a new

shall

definition of a transcendental expansion.

not an algebraic expansion shall be called a transcendental expansion. We find it necessary first to reconsider the and let 8 be a system expression "algebraic dependency." Let L>

Every expansion which

is

K

a be an element from L. Now 8 determines an expansion K(8) of for which ($) ^ L. If now a is algebraic with respect to K(8\ then we say that a is ALGEBRAICALLY DEPENDENT upon IS. Therefore a satisfies an equation with coefficients which are rational functions of the elements of S with coefficients from K. Further, a system T of elements a from L is said to be algebraically dependent upon a system 8 from L if every element a of T is alge-

and

of elements from L,

a

If

K

dependent on

braically

let

K^ K

S. In this case

algebraically dependent on

is

A

Tis

entirely contained in JT($).

8 and

A n -ia n ~

+ Aia+ ...+

l

+ an =

expresses this dependence, then the elements At belong to 7T($) and are therefore rational functions of a certain finite number of elements

That

s1} ...,#.

K(slt

...

,

.
these

is,

A, are already contained in a finite sub-system of 8, Hence we

coefficients

= K(S'), where 8'

is

t

have

THEOREM

If a

1.

algebraically dependent on S, then there exists

is

finite sub-system 8' of 8 such that a

THEOREM

2.

T and S to If

$

s

is

braically dependent on

Let

K(8 } = K l

KI ($2)

algebraically dependent on

a

8'.

from this theorem that when we are considering the notion T being dependent upon another system S, we may

It is seen

of one system always consider

that

is

l

is

and

be

finite collections of elements.

&

2 is algealgebraically dependent on S3 and then 83 is algebraically dependent upon Si. ,

8l}

K(SJ = K*. From

algebraically dependent on

K K

34,

KI and

Theorem

K ($

3)

is

5,

we know

algebraically

K

< L ($2 ), hence the elements of 2 ($8 ) are 2 dependent on K%. Now with to K^ ($2 ) and therefore with respect to K^ by algebraic respect Thus the elements of 2 (Ss ) are algebraic with 34, Theorem 6.

K

respect to

K(S^) and the elements

of

$

8

are algebraic with respect to

$3 is algebraically dependent on Slt and 2 is alge2 algebraically dependent on braically dependent on $1} then Si and $2 are said to be EQUIVALENT JT(/$i).

It follows that

If the system

*

$

is

$

,

Steinitz, Algebraisclie Ttieorie der Korper, pp. 114-119.

$

MODULAR INVARIANTS

61

[N.B. The reader must be careful to distinguish between equivalent systems and equivalent expansions.] If Si and SQ are equivalent systems, we write Si ~ S.2 From Theorem 2, it is seen that if

systems.

.

Y

AS X

~ $j and

A

S

z

~ Ss then

SL

,

~

/S.

S is said to be ALGEBRAICALLY REDUCIBLE with respect to S contains a single element which is algebraic with respect if S is equivalent to an actual sub-system of itself. If S is not

system

K either to

if

K,

or

if

A

reducible system algebraically reducible it is said to be IRREDUCIBLE. contains at least one element that is algebraically dependent upon

S

the others.

This follows from Theorem

THEOREM

1.

Every sub-system S' of an irreducible system

3.

S

is

irreducible.

THEOREM which

A

Every reducible system

4.

S contains

a finite sub-system S'

is reducible.

transcendental expansion

is

said to be PURE

the adjoining of an irreducible system.

Let

if it is

S = $i

-..,#

t

obtained by be a finite

irreducible system, then L-K(xi ...,# M ) is the field of all rational is equifunctions of n unknowns. If S be infinite we can show that valent to K(&i, r2 ...) where there are an infinity of #'s within the y

L

,

bracket.

Let

/Si

= x^

.

xn and

.

.

,

$ = #1, 2

,

y n be two

finite irreducible

systems with the same number of elements.

THEOREMS.

K'($) N

equivalent to

is

and every element of the following manner

is

K (#

That

a ).

self-corresponding.

We

is,

K ($) -

can obtain

:

arid similarly

It

K

has been shown in

K

K

33 that K(x^) This theorem

(y^)

and

so

by an ^-fold proved

for the

are infinite systems. Suppose now that the system S is irreducible with respect to that a is an additional element.

K and

application,

(S-?)

case where Si and

THEOREM

6.

(S^).

may

also be

S

2

If

S

irreducible but

is

S+a

is reducible,

then

a

is

algebraically dependent upon S.

From Theorem

T must

4,

contain a.

S+ a Let

contains a finite reducible sub-system jTand + a. Now T' is a finite sub-system of

T=

T

MODULAR INVARIANTS

62

S

and

is

Suppose that jT'=#i, ...,#* and

therefore irreducible.

K =K

let

= K(T') where every field IT, is a pure transcendental expansion of v -.^. Now, if v (a) were transcendental with respect to v 32, Theorem 2, every element of T then, from

K! = ^T(^i),

,

n

n -i (sc^)

K

t

K

K

,

would be transcendental with respect to the n remaining elements of T j i.e. T would be irreducible which is a contradiction. Therefore n (a) is algebraic with respect to n so that a is algebraic with respect to n and therefore with respect to T' and finally with respect to S. :

K

K

K

L>

THEOREM 7. Every system S of elements from K, of which not every element a is algebraic with respect to K, is equivalent to an irreducible sub-system S' ^ 8.

We may suppose that S contains no elements which are algebraic with respect to for otherwise we can neglect these. Thus every a of L is transcendental with respect to K. If S is a finite system the

K

t

theorem where S

is

is

We

obvious.

an

shall not require this

theorem

for the case

infinite system, so for the proof of this case

we

shall

refer the reader to Steinitz*.

If L~K(S} is a transcendental expansion of K, then from the theorem just proved 8~ S' where S' is irreducible. Thus every element of j$ is algebraic with respect to K(8'\ and K(S} is an algebraic

Hence

expansion of K(8'}.

THEOREM

8.

follows

Every transcendental expansion is obtained by an a pure transcendental expansion.

algebraic expansion following on

Let

L = K(8}

where

8

is

equivalent to the irreducible

8'.

If the

system 8' has n elements, then n is said to be the TRANSCENDENTAL DEGREE of L with respect to K. If 8' contains no elements, i.e. if L is algebraic with respect to K, then we say that the transcendental degree is n } with respect to is zero. If the transcendental degree of of

L

K

L

L

n elements

at least one irreducible system of but no irreducible system of more than n elements.

then there exists in

36. Rational Basis

Theorem

THEOREM f. Jf {/}

a

of n indeterminate^

is

of E. Noether.

collection

with

of rational functions

coefficients from

f(xlt

. .

.

,

#n )

afield K, then from {f} it is possible to choose a finite number of functions /i, ... 34/m such that every is a rational function of the f^^ *..,fm with coefficients from #1, ...,#

f

the field

K. * LOG.

cit.

t Gott. Nach. 1926, Heft

1, p. 28.

MODULAR INVARIANTS Such a system of functions flt The above theorem is trivial if {/}

...,/m

63

called a

is

RATIONAL

BASIS.

a finite collection.

is

I. Now jfT(d?], &n) is a pure transcendental expansion of JTand of transcendental degree n. The total of all the rational functions of the/'s from {f} with coefficients from IT also gives a transcendental ,

is

L = K({f}}.

expansion of K, namely

Obviously

K
!/h

>

L

of

^

then

is

S of

he an irreducible system

L.

and ^

1

If it

n.

Now

let

L

can be shown that

a finite and therefore an algebraic expansion of H=K(yi, ...,#&), is a finite expansion of ff, then .then the theorem is proved, for if will L 34 ; i.e. L is obtained by the (Li, ... LI) from Theorem 4, is

L

H

,

L

from I/ to H. 1) ..,, L adjunction of a finite number of elements l/ last that This shows Hence h 1/j ly y ). every element

L^K(y

from T/],

Z/

..,#/

.

.

.

,

,

,

. .

.

,

t

?

and therefore every / from {/} is a rational function of l/i, >"\Li with coefficients from A, and since the ^, ...,^

are also elements of

1>, i.e.

are also rational functions of the /'s from

{y }, therefore those /'s which are used

for the

T/'S

and the 1/s form a

rational basis for {/}. II.

It

L be

K
remains therefore to prove that if K(tri, ... #n ) and h, less than or equal to n with respect S y^ ..., y/ then I/ is a finite alge,

of transcendental degree to Jf with an irreducible system if

braic expansion of H^-K(y transcendental degree as K(a^ l

...,

,

.

^

t

r

. .

,

,

).

tt ).

If A

=

then

17" is

of the

same

Since ^ is an irreducible system of

xn ) also, hence every element of K(x L and therefore of K (X # ), and thus ^ itself, is now algebraic with respect to H, since every system yl9 ...,#,# is reducible with respect to A*". It follows that every element a? w ) also is of L and of K(x ly algebraically dependent on H. Hence L and K(x^ ..., x^) are algebraic expansions of IL Now since . .

,

.

.

,

} ,

. . .

,

ft

. .

,

then

K(XD

number of

...,

?

n ) is

obtained from

H by the

xn and

adjunction of a finite

by Theorem 4, is H. But since every element of also an element of L therefore ^L^ #w ). (x Hence I/ also is a finite algebraic expansion of H. The Rational Basis Theorem has now been proved for the case where h - n. Suppose next that h < n. S is an irreducible system in L and in K(XI, ..., #). The suffixes of the #'s can so be chosen that algebraic elements

^,

...

34, a finite algebraic expansion of 3

H

,

K

therefore

is

H

. . .

,

,

MODULAR, IOTABIANTS

64 is

-

an irreducible system of K(x^ ,. d(yi, y*, a?i .-,#<)

T Jacobian

--^ ,

o

(^

.

.

m ---^ ,

,

#

...,

,

,

would the

for otherwise

tt ),

.

,

.

vanish for

all

.

t h+1

,

,

?n

...,

,

,

so that

#,J

the #'s would not form an irreducible system of K(%i, #w ). The xn is therefore irreducible with respect to y system JT= ^/i+ i yh and hence with respect to II. We may suppose then that is an irre. .

.

,

,

. . .

. .

,

.

,

X

,

ducible system with respect to L also for otherwise #VH, for example, would be algebraically dependent upon L, i.e. on certain elements from L, so that # +i would be algebraic with respect to y ly ... yh since every element of L is algebraic with respect to /S Hence Jf is algebraically independent or irreducible with respect to $, II and L and hence with and L, where between respect to every medial field ;

fc

,

,

y

.

y

M

H

It follows that

K=

,#) and

-n,

are pure transcendental expansions of 37, is therefore irreducible with respect to j/f

Z and JT respectively and $ were not the case

for if this

:

K

then we should have an algebraic equation with coefficients from n would not then be relating the y^ ..., ?//,; and y l9 >->'yh> &h+i t

an irreducible system of A"(r1?

K to

K

fore

is

L

. . .

,

We

#n ).

...,

of JT as the coefficient field.

Since

>

can now choose

K<-L
...,

^ w )- But the transcendental degree

^ and that of and K(x^

K(x^

. . .

,

...

#/t ) or

,

7//

with respect to

)

K

and

.//

. . .

(,r1}

K

#)

,

K in place

#n ),

therefore

of jk with respect is

also k.

There-

are of the

same

transcendental degree h with respect to K. But the theorem has been = proved for the case where Ji n, so that L is a finite algebraic expansion of

K with a rational basis y

l

yh

. . .

,

,

L = S(yl9 We

now

9

^ yh

,

,

. . .

,

zk

,

say.

^ ...,^)

Thus

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

(361).

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

(36*2).

consider the field

M=K(y We may

...

,

l9

suppose here that ylt

...,y ...

,

y/

fc

,i, ^ 1}

...,^ ...

,

^

fc

)

are from Z.

K

therefore be chosen belonging to L.

yL9

...

,

^

K and therefore

certainly are, also i> is a finite Algebraic expansion of which do not belong to h + k elements are adjoined to

K. They can

Obviously

H<M^L

........................ (36-3).

Formulae (36*1) _and (36*2) give M=L, so that L thus L < M. It follows from (36 '3) that

is

contained in

M for

L < Z,

M

........................ (36-4).

MODULAR INVARIANTS

65

M

already, is Now every element of M, which is not Contained in is a pure transcendental transcendental with respect to M, since is which does riot belong to expansion of M. Every element of is algebraic transcendental with respect to M. Also every element of with respect to y ly ..., 2fa and therefore with respect to M, so that

M

M

L

L

ML = of

We an

is

K(jyi)

-, 2//u

IL The theorem

for

1,

,

s&),

of E. Noether

and

I/ is a finite algebraic expansion therefore proved for all cases.

is

K
have also the corollary*. If

M

...,#) and

l ,

be a finite expansion of algebraic expansion of L, then must has a finite rational basis by the last theorem.

M

M

L,

'

The

37.

Fields

KJ>f

.

K

{p) is root of an element of a unique, p p were two/>th roots of a, let them be a and b. Hence a = a = b p p b - (a and therefore = a by, whence a = b. We can write the pih

The operation of finding thejoth

for if there

root of

If a belongs to

uniquely as of.

K

may

it

or

may

not happen

i

that OP also belongs to not happen.

Let

K.

We

shall consider a case

where

this does

L = K(n) be a simple transcendental expansion of a field K with L also must be of characteristic p. Let a be an eleof K and of grade k [the grade of a is maximum not of L but

characteristic p.

ment

a =--f(ji)lg (/*.)]. It can be shown that (degree of/; degree of g) where The equation is of grade Jip, which is certainly greater than unity. a1 = would then reduce to an equation in /x, with coefficients from a?

K

'

fji

so that

/A

would be algebraic with respect to K. This

is

impossible, i

i

^

L

does not con tain p p is said to be PERFECT f if every polynomial F(a) with field coefficients in the field and with repeated linear factors is reducible in irreduis said to be IMPERFECT if there exist in the field. A field

hence the equation a

K

A

is

impossible, that

is,

.

K

K

cible polynomials with repeated zeros in a super-field of

Obviously every characteristic

K.

field of characteristic zero is perfect.

p we have the

following theorem

There are perfect and imperfect

K

A

K

For

fields of

:

(p)

1

perfect if a** belongs (v) is to imperfect. provided that a belongs to K. In every other case (p] is from K, then from that that is a and then y Suppose

K

*

R

31,

's.

is

K

K

K

Theorems,

(

^__

Gdtt. Nach., loc. cit. p. 31.

y

t Steinitz,

loc. cit. p. 50.

5

MODULAR INVARIANTS

DO

K

We

8 is either from now or from an algebraic expansion of K. pj"= (x~ 8) pr y. Suppose then that ^ try to find irreducible factors < of x = xpr - <^r (0 ^ r ^ /) be an irreducible factor. can only have r -

where

We

if B

= yp ~f

belongs to

K. Otherwise

>

r

which case we have an and hence (p) is imperfect.

in

0,

K

irreducible polynomial withj/ repeated roots, i

K

p

and that/0^) is Suppose now that both y and y belong to the and which has repeated a polynomial which has coefficients from the zeros. Then is (/,/') of at least the first degree in x. If/' (x) 4=0, then (p)

K^

the degree of (/,/') fore reducible.

than the degree of /(#), and /(a?)

is less

If/' (x)

= 0, then

/ (#) = #

+ $!#p +

,

#i

p

is

therefore reducible

:

are

21

,

/Or) = (> and

p

suppose that

ap

K

elements of

all

+ ajP a? +

hence

This leads us to consider the

We

(vp

n).

i

i

But by supposition a n p

-

+ a v x vp

...

(p]

fields

.

.

.

jftT

(p) ,

so that

i + aj* xv ) p

in this case is a perfect field.

which we

shall designate

K has the characteristic p, then

bp

= (a

there-

is

&)*

;

ap b p = (aV) p

;

a*

:

bp

= (a

:

by

Kpf

.

fe^.

K constitute K which we not the denote by K*. (N.B. K K and are seen to be isomorphic, or K~K same as K obvious also that K ^ K.

We

see therefore that the ptli powers of the elements of

a sub-field of

shall

p

is

P

jST*

.)

.

It is

p

If is

we now add

an element of

a? as a new element whenever & we obtain a field X such that K= L p

to A" every element

K but not of

J5TP ,

.

i

L

=

K

p

We

can repeat either of the above For this reason we write p *. Such fields as processes as often as we wish and obtain the fields and we shall call them RADICAL fields of K. K*~f are expansions of .

K

K

In the same way we can obtain radical rings. If L be a radical field of K, m then there must be a minimum exponent m such that the p -\h power of every element in L is an element of K. In this case m is called the EXPONENT of L. Similarly we define the exponent / of an element a of L. It is obvious that/^ m.

38. Expansions of the First and Second Sorts. We can divide algebraic expansions into two sorts*. Let g(x) be a polynomial which is irreducible in the field K. We say that g (x) is of the FIRST SORT

if it

has only simple zeros when considered in an expan*

Steinitz, Zoc. cit. p. 62.

MODULAR INVARIANTS

67

Now let the element y be algebraic with respect to K] then an equation of the type g (x) = which is irreducible in K. If g (x) is of the first sort, then we say that y is an ALGEBRAIC ELEMENT OF THE FIRST SORT with respect to K. An algebraic expansion L of is said to be of the first sort, if every element of L is an algebraic sion of

y

K.

satisfies

K

element of the

WQ

sort with respect to

first

K.

see that every algebraic element with respect to a perfect field

K

is

of

K^

and is imperfect. Suppose is a Suppose then that also that g (x) is a polynomial which is irreducible in K. We have seen the

first sort.

that there

is

associated with an element a from the radical fields K**,

1

K^

~x

an exponent/, such that ap} lies in JTbut ap/ does not. We extend this notion to the case where a does not actually belong to a ',

...

radical field, as follows

:

= G (x vf ), f > 0, i.e. g (x) is a polynomial in x pf for some maximum/; or/ = and g (x) is not identically zero and has thus no common factor with g (x\ so that g (x) = has only simple zeros and is f therefore of the first sort. If /> 0, g (as) is the p -th power of a polyEither g (x)

K and not of the Let polynomial be h called the exponent of h (x\ the p -th power of h (x) f ~th belongs to K while the p power does An algebraic expansion of K said to be of the SECOND SORT nomial in

I>j

In this case /

this

first sort.

is

~l

(x)

f

for

is

not.

if it is

is

not of the

first sort.

THEOREM. first sort

further sort

L is

Tf

then there exists

with respect

L

is the

with respect

an

algebraic expansion of

a medial

field

Z

,

i.e.

K and such

to

K^ L L

that

is

aggregate of all the elements of to

an imperfect

^ L,

such that

L

field Q

is

K,

of the

a radical field of L L which are of the first .

K.

This theorem holds for infinite algebraic expansions, but it is only Let us call L the system of all first sort with respect to K; then

proved here for the finite case. elements in L which are of the

K

K

= (LJ) (LJ), where L9 is obtained by removing from LQ element which is also contained in K. Further, by Abel's Theorem* every on primitive elements, a single element y can be found such that obviously

K (Ld) = K

(y),

'

where y

is

K. Let a be any element [JT(a)

algebraic and of the of ^(y) and let

:*] = *

,,

\K(y)

:

first sort

K(a)] =

Steinitz, loc. cit. p. 52.

,,

with respect to

MODULAR INVARIANTS

68

Hence m^m^ = n l n z = [K(y) therefore

'

K~\ and since

K (a?)

is

a sub-field of

m 2 ^n 2 Wj^Wj, ^W 2 + ha:*- 1 + ... + b mz function which vanishes when x - y, then

K

(a),

.

Now

let

be the irreducible

vanish for ./f

);

a = y p and

2

n 2 = \K(y) 2 and Wj

Hence K(y)

sort.

K(Lo)

and LQ

b

r

^

belong to the

field

K(o?)} = [JT (/) A>)] S 2 p MJ, that is, a and a are of the same degree Since this is so, [J5T(a); ^T(o p )] = 0. But, by a :

:

with respect to K. theorem of Steinitz*, [K(a)-y K(a p )] -p than zero. It follows that the exponent that

...,

therefore

w -%

Hence

the coefficients b^,

will

is

is

an expansion of the

contains elements of the

a medial

field

between

.

if

the exponent of a

is

zero

and

It

first sort.

first sort

K and L or

is

greater

so a is of the first

has been proved

only, hence

Now if S be an element of L of exponent / with respect to K, then vf g (si) = G (x ) = G (y), say, arid G (y) is certainly irreduciWe in K. Thus G (y) has only simple roots, so that y - a pf is of the first sort and therefore an element of L i.e. every element of L is a radical element of L and therefore L is a radical field of LQ It is easy to show that ,

.

uniquely determined by L. If L is a finite algebraic expansion of an imperfect field K, then is i, ..., 8 m ). Suppose./^- is the exponent of 8 and that the maxi-

I/Q is

t

.

mum /J

is/,

then/ is the exponent

of the expansion of

39.

The Theorem on Divisor Chains f.

Let

E be a

commutative ring with elements a

element and with no divisors of

We

zero.

t

L and L ^ LQ&. .

b, c,

.

.

,

with a unit

shall consider in particular

sub-rings a of Ji which have the following properties (1) if a and b belong to a, so do a + b and a b\ (2) if a belongs to a and r belongs to R, then ra belongs to a. Thus a is the total of all elements such as :

r'a

+ r" b +

r"'c

+

...

and

is

said to be

a = ($,

An

ideal

tained in

a

is

a.

an IDEAL

b, c,

in

R.

We

fo if every element b of fc is conThis idea of division is explained

said to divide the ideal

This *

f

is

written 6

^

a.

write

...).

Algebrauche Theorie der Korper, p. 63. Van der Waerden, Moderne Algebra, vol. n,

p. 25.

MODULAR INVARIANTS

69

by van der Waerden*. With this definition of division the f the ideals (a, t), C, ...)> and L.C.M., written [a, ft, C, ...]> = a said to are An is be FINITE obtained. ideal a 2 M ft, t, a, (a ) if n is finite. a 1} $ 2 ... a are said to form a BASIS of the ideal a. We shall only consider rings in which every ideal is finite, and we shall now prove the following theorem. Every ideal of a ring has a finite where &i+1 basis if, and only if, there exists no chain of ideals a < a 2 <

more

fully

written

H.C.F.,

.

l

,

,

.

,

,

. .

,

,

t

,

an actual divisor of a,, which chain does not come to an end after a finite number of steps. This is known as the theorem of divisor chains. The proof is as follows. is

We

shall first suppose that there exists

no

aa < a 2 < a 3 <

. .

infinite chain of ideals .

,

an actual divisor of a > Now if a be an ideal without a finite basis, then let x be an element of a. Since a =t (a-^), then a must contain an element a s such that &a'=(<&i,# 2 ) is an actual divisor of a/ = (i). Since a4~(X, 2 ), there must exist an a s such that where a

+1 is

7

a 8 = (i '

is

an actual divisor of a 2 and so ',

2ia)

}

on.

We would thus

obtain an infinite

chain of ideals in contradiction to hypothesis we were therefore in error in supposing that an ideal existed without a finite basis. :

Suppose now that every ideal has a

finite basis

a x < a2 < a s < be an

Let ft = (3i, a 2 itself an ideal for

Now

ft

is

)

,

;

if

be the

a and

is

let

an actual divisor of the

H.C.F. of all

b are

and

-

such that each

infinite chain of ideals

preceding.

-

the ideals & lt a 2)

two elements of "D, then suppose

a^ and that b belongs to a/. If ^be the greater of i and j then both a and b must lie in a# and hence a + b, a-b and ra belong to & N where r is an element from R. It follows that a + b,a-b and ra lie in ft, hence ft is an ideal and by hypothesis will have a finite basis di, ...,d Each d, must belong to some ideal a^t Let the v be then M; every element of the basis belongs to &M and greatest

that a belongs to t

,

.

.

ft

t

ft belongs to a^ ; that is ft ^ Hjf the ideals a -, ft > &M It follows that ft

hence

.

z

be shown that

ft

= &M+t where

&M ~ and the

a^

.

But, since

= a^ and in

ft

a positive integer.

fljkT+i

= #M+2 =

is

the H.C.F. of all

the same

way

it

may

Hence

j

not an infinite one, since a^+t is not an actual This concludes the proof of the theorem.

divisor chain

divisor of

t is

.

is

*

Loc.

cit. p.

29.

5-3

MODULAR INVARIANTS

70 40. R-Modules. Let

8 be a super-ring of R,

where both

S and R are commutative rings

with a unit element and with no divisors of of

& which 1.

contain

If a

and

/3

Consider systems

zero.

R and which have the following properties do a +

so

M,

"belong to

/?

and a -

M

:

/3.

M

M

and r to R, then ra belongs to M\ thus 2. If a belongs to is the aggregate of elements r'a + r" ft + ... of $, where a, /?, ... are from $ of the above sort is called an and r', r", ... are from R. system

A

R-MODULE.

a, /J, ...

number of elements, then we say that If every element of a

we say

that

M

and a module

is

j?V

M M

form the BASIS of If and

module

a

the basis has a finite

if

finite

jR-module.

Mis also contained in a module N, then

by N.

divisible

is

has a basis vl5

If ...

,

a module vn

,

.ftf

has a basis

/*!,

...

,

p,

m

then the module with the basis

M

v i g called the product of and JV^ and is written /** /*
vi

M

JIfZV.

,

,

. . .

,

M

//.

,

M

. . .

j

therefore in

A

i}

of the lengths 1, 2, ...,#, e.g. &,...,&. Now of length ^ i also form an 72-module, the elements in and every element in t is of the form

A A

all

+

!ifl

The system

of elements a w

obtain in this

If

,

M&+

all

manner a sequence

from

+& R

y

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

form an ideal

a,

let

us suppose that

Van

R.

We

of ideals

...

= a, + i = ...=o.

B is an actual divisor of A,

so that every element *

(401).

a* in

A contains no element of length/, then, since A$ = Aj+1 = Now

t =1=

,

M elements

the system of

namely

-

,

der Waerden,

loc. cit. p. 87.

then

= 0, must

MODULAR INVARIANTS

A K is

of

since

contained in B, since

it is

at most of length

X.

71

A K ^ A
similarly

a A ^i) A

B

EK

also contained in

(40-2).

B

an actual divisor of A, there must exist in one or more elements which are not contained in A. Let ft = b^ + ... + b t t be such an element whose length i is a minimum. If ft t were not an actual Since

is

divisor of a*, then every

a

of 6* would be contained in a*

bj

and

= 0i,^i+

+0-i,t&-i+&i< would be an element of and therefore of A also. Now if a is in J5 so is y = /3 - a = (&,_ - a M ) fi + ... + (&;_! - i_ M ) &_i, but y cannot be contained in A for otherwise /3 would be contained in A which is contrary to supposition. Hence y is an element of B but not of A and is of length ^i 1, which is also contrary to supposition ; hence ^ is an actual divisor of a^ We have shown therefore that if A
^

f ,

some minimum value of i. Suppose now A < A ^ < A ^< and suppose that the theorem of (l)

(

then associated with

From equation

x)

af

and

,

we have

for every

aA w^a*<

2

< a2

X=

(r)

<

1. 2,

... ...

< &k (r) ,

finite one.

rA be r

finite chain.

,

r +1)

aA

(1)

F

=

R and if every ideal in It is finite,

...

for all

a Aro+1) (

{3)

<

...

and is

a

a finite then every R-module in

This theorem can be stated as follows

S has a finite

t

Let the last member

ro) =-

so that a A

A^
super-ring of

R

.

k that

valuesofX. But since

we have a

of modules,

Ua/u....

must be a

maximum

we have a chain

divisor chains holds for the ring

are the ideals a/*

this last

let the

that

i.e.

. . .

(r ^

(40 '2)

But by hypothesis be

A

(

:

If

JS is

basis.

R is an actual sub-ring of F, and let a be an element of We can therefore form a super-ring of R containing all + rm aw where m is arbitrary. in F of the type r + r a +

Suppose that but not of R.

the elements

...

a

,

Here we have the 72-modules 2 1 l ^i = (a',a ), A 2 = (a', a a ) ...(40'3), This chain is a divisor chain since A ^ A i+ If for a certain m, A m = -4 TO -i, then there must be an equation of the

^o = (a) = W=22,

,

following type

ftm+

^^

+

,

i

t

...

+n =

(4(, 4)j

with integral coefficients r ..., rTO _!, and therefore every polynomial of m~i. In this case the super-ring degree m+jt? in a is contained in ,

A

MODULAR INVARIANTS

72 has a

1 basis a, a ,

module

finite

equation as (40 '4) we say that it or shortly that it is R-ENTIRE*.

a"1

-1 .

When

a

such an

satisfies

entirely algebraic with respect to

is

R,

definition an ^-entire element a defines a finite super-ring of

From

R which

we

THEOREM

S

by H a We mean by this that E a is we do not mean that R a is itself a finite ring.

shall denote

R

with respect to

Let

...,

:

S

Every element of a finite super-ring

1.

R

finite super-ring of

be a

finite

.

R<8

that

so

of

R is entire.

and

a be an

let

S.

,

t

a
Hence, using that

S in the

to say a

is

=

+ r iz
a 01

r,

+ r.m
...

Kronecker sense,

\rlk

.

-<*&k

=0

n or a

+

...

\

-0;

is entire.

THEOREM 2. Jf 8 is a super -ring of R and if every element of 8 and if a from T, a super -ring of 8, be 8-entire, then a also

R-entire

is is

R-entire.

Since a

from

is

$ and

m ^entire, then a + is

o-

m ^a

therefore jft-entire.

m~l +

Hence

...

+

a-

=

0,

where every

R

obtained by adjoining the elements the super-ring S of R. Since every o- 4 is ^-entire, S is a finite super-ring of

is

R

;

,

...

m_T

cr ,

to

also a is $-

and

^a

theorem. If every ^-entire element of the quotient field is ENTIRELY CLOSED. an element of R, then we say that

last is

is

are finite super-rings of 8 and 8 respectively. therefore a finite super-ring of jtf, so that a is 72-entire by the

8a

entire, so that

8a

or,

a is also entire with respect to

P

of

R

R

THEOREM

If any element of R can be resolved into prime factors

3.

uniquely, then

R is entirely closed.

For, let a be

an ^-entire element of P, then since a is ^-entire + rm = am + TI a "- 1 + (n from R). 1

.

.

Since a belongs to P, then a =

and

5 divides

an element of *

Van

theorie,

.

*

,

t

and

m and therefore divides

R itself.

That

is

to say,

s

.

being elements of ^. Thus

Let

#

= s^, then

a=

=^

t

,

5

R is entirely closed.

der Waerden, Moderne Algebra, Teil 2, pp. 88-89, and Landau, Zahlen-

Bd. 3, p. 32.

MODULAR INVARIANTS

73

A Theorem of Artin and of van der Waerden. Suppose that R is a commutative ring with a unit element

41.

with no divisors of

in this ring is finite

applies; (2) that

We

zero.

R

and that therefore the theorem of divisor chains entirely closed in its quotient field

is

R has a characteristic p > 0,

if

and

shall further suppose (1) that every ideal

then

is

the radical -ring

P\

(3) that

R p finite with R is a field R is a finite

respect to R. The third of these conditions is satisfied if which is either finite or perfect, or still more generally if algebraic or a transcendental expansion of a finite or perfect

be

with respect to R, so also

finite

is

R

1

**

If

Rp We

with respect to R. Let Plea quotient field of

finite

now prove van der Waerden's Theorem *.

shall

field.

i_

IS a finite expansion of P, 8 the ring of all R-entire element* of 2 then if conditions (1), (2) and (3) obtain for R, they also obtain for S. shall first prove that S is entirely closed. This means that if cr from 2 is entire with respect to S, then or is an element of $ itself. Now m m ~ l + ... + s if o- is entirely algebraic with respect to S, then v + s\
H,

;

We

Hence S is entirely closed. must now show that S

S.

We

is

would then hold

divisor chains

a finite ^-module, for the theorem of Also S would be finite with

for S.

i

R and so, following the isomorphism, S v would be finite with L L 1 to R p and R v is finite with respect to R, so that S p would be

respect to respect

,

i

R

with respect to and hence with respect to S, since R <JS <S P and hence condition (3) would be satisfied. We have therefore still to finite

$ is

show that

Now S

;

a finite 72-module.

a finite and therefore an algebraic expansion of P. The most general finite expansion of is obtained by an expansion T of the is

P

sort followed

first

of S,

r

<$

i.e.

maximum

e

S^r

by finding a radical

*'*.

Now

let

field of T.

C be

the ring of

all

C pe

be the exponent

the .^-entire elements in r,

i

then

If e

of all the exponents of the elements of 2, then

_i

is

the ring of

all

^-entire elements in

r P*

since an element of

\_

r pe

is

entire

if,

and only

if,

its

e

p th power

is entire, i.e. if its

j_

is

in C.

Hence

C *

Gottinger Nachrichten, Heft

1, p.

26 (1926).

e

p th power

MODULAR INVARIANTS

74

Now F is a finite expansion of the first sort of the field P, thus r = P (a).. Every element y of T and therefore every element of can be written as

where each

y=p

p, is

4-

Pitt 4-

p2 a

from P. Since r

fore the equation

which defined

2

and solving these ll

Now and

is

n

for

"

+p

+

+ p2
=~po

1

a*

^M~l

-v/

r,^

+1

ft

W

,

7t\

,

so that

^_

P-iC

~M

(

we have n

v== l>2, ...,),

-

.

D

and

is

therefore also an element

Hence 7=

y,

(

+ Tja +

To

Every element y belonging to that

aw

...,

,

D

Hence

= belongs to Jf. Also T M /)p M since it is an integral function of the a's of

first sort, there-

product of the differences of the roots is an entire element of P, i.e. that is, it belongs to P; also it is entire

Z) is the square of the

therefore rational.

,

a,

equations y,,

1

p n -iQ-

an expansion of the

is

Let these be a j} a2

has n different roots.

n

... 4-

4-

C

+ Tn _! a *" 7

. .

.

1

).

can be represented in this manner, so

C is a module contained in the finite module M- -P\~i\i~i\>

Since A/

is

a

finite /^-module, KO is

Y

6

,

R ^ C ^ M. Now

for

L with respect to

Jt,

and hence

isomorphism,

/r

)

is finite J

fl

with respect to

(7^ is finite

R*

e \

also

by

i

i

Jt

~

pe is

finite

with respect to

J?,

so that (7^

e

must be

finite

i

with respect to JR; but C ^ /S ^ (7^. Hence ^ is finite with respect to JR and is an 7?-module. This completes the proof of van der Waerden's

Theorem.

42.

The Piniteness Criterion of E. Noether*.

P be a given

P

# w ] be the polynomial ring field and let [.^ tfn with of rational integral polynomials in the coefificients be a division ring whose elements are belonging to P. Further, let Let

,

.

.

.

,

^

.

,

.

.

,

J

*

Q'ottinger Nackrichten (1926), p, 31.

MODULAR INVARIANTS

P polynomials such that P<J< P [x^ to

P

/i,

72

whenever there ,

,

I*, so that

is

75

J is finite with respect n J a finite number of elements of / can be expressed as a rational ...

,

sr

].

contained in

every element

integral function, with coefficients from

P, of

this basis

Il9

...,

Ih9

in

Pp

is

which case

We

shall suppose also that if

has the characteristic p, then

with respect to P-

finite

E.

P

NOETHER'S THEOREM. A ring J of polynomials in x from P, which has no divisors of zero, is finite with .

l ,

.

coefficients

P

if,

and

respect to

only //*, tlwre exists within J a sub-ring R which P, suck that every element of J is If -entire.

This condition can put R=^J. It JT is the quotient is

is

is

.

,

xn with

respect to

is finite

with

if /is finite with respect to P, then we to be sufficient as follows. Suppose that

necessary; for also

shown

field of I? cund

that

L

is

the quotient field of J. This

possible since the rings have no divisors of zero.

Hence

P
P (x)

where

is

the field of

all rational

functions of

x with

coefficients

from P. We now make use of tho corollary given at the end of 36. It can be demonstrated readily that L is an algebraic expansion of K^ and hence L is a finite expansion of K, where, if P be a with p > 0, we understand the most general expansion of the first or second sort.

P^

Call

$ the ring of

/^-entire elements of L. Since therefore every

./ is /f-entire,

J is

element

a sub-ring of $, i.e. J < /S < L. is finite with Suppose now that J? be entirely closed. Since respect to P, then 72 = /*[*/! (,r), ..., g T (w)~], where g (x} is a polynomial in of

y

R

t

#!,

...

,

xn with

element of J. applies

and we

P

R

and belongs to and is therefore an a finite ring the theorem of divisor chains can also use van der Waerden'.s Theorem. is a finite

coefficients

Since

from

R is

R

K

is its quotient field. L is a finite expansion of K\ 8 is ring and the ring of 72-entire elements of L. The theorem of divisor chains for

72-inodules therefore holds for $.

module

J is

an //-module and has therefore

h s (x\

J

i.e. can be every f(x) of written *&f(x) - AI h+ (x) + ... 4e hs (x), where the A's belong to R. The A' a are therefore polynomials in the
a

finite

basis fh(x\

...,

A

together form a finite module basis of K. If be not entirely closed, then we must

first construct a sub-ring R T of R which is finite with respect to P and which is entirely closed and

76

MODXJLAB INVARIANTS

R

with respect to which using T in place of R,

We can proceed

is entire.

We

with the above proof,

refer the reader to E. Noether's

paper for

this case.

Theorem

43. Application of E. Noether's

to

Modular Co-

variants*. Consider the group r of all homogeneous linear substitutions with A h be the different from a Galois field, and let A2

A

coefficients

substitutions whore

A

}

,

,

formation Ait transform the set of variables sr^,

...

^

,

,

Now

the identity substitution.

is

l

,r

1

,

let the trans-

...,a? m

to the set

Consider also the resolvent of Galois,

where a < h and a +

a,

+

... 4-

am

= h.

in,

Let 0*=

2$

1=1

so that

,

/rM. l

We

can choose the w/s so that

#;t

^0, unless

7r

-,/,

any symmetrical function of 19 ..., h must be an absolute cois such a function and hence the coefficients Uaai am (%) )

variant. <(s,

...

are absolute covariants.

Let from

J

all

be the ring of the elements

every element

/ of ./

absolute covariants and

all

Uao

.i

is

...

a m (?*)

so that R,

^

J.

R the ring obtained If

we can show that

then from the theorem of the previous

72-entire,

i

J

will have a finite basis provided as before that paragraph, finite with respect to P.

Now

if

we put

z

=

?A

x^

4- ...

(^.r^+.-.+^^^+s^.,.. for all

w

t 's.

Suppose then that

,rN S-rf

then

where a +

a,

= h and a <

//.

+ um

= 8J

iij

^

fl0

..

1) 1

is

then

.r^/,

w(

a

Pp

^i

+-+^i>r - r = o 1

(using S in the Kronecker sense),

Oa

.

..

(,r)

=0,

This last equation shows that 3\

R

is

entirely

and therefore with respect to algebraic with respect to a portion of ft itself. It follows that every polynomial in the #'s and therefore every element of J is entire with respect to /?. *

GMinger Nachrichten

(1926), p. 33.

MODULAR INVARIANTS

77

:i

P

all

P

and the GF[p*], then P* is finite with respect to the conditions are satisfied and the theorem is proved for the case

If the field

is

The extension is made to covariauts involving forms by regarding the coefficients as additional

of universal covariants. coefficients of the

variables.

The proof

also holds for relative covariants as well as for absolute,

P. The proofs also hold if n \ so that we have proved in case the theory of modular coevery

since the factor appearing also belongs to the #'s and a's are unknowns of the

GF[p

the finiteness of covariants for variants.

APPENDIX Dickson in his History of

I

the

Theory of Numbers, vol. 3, chap. 19, the papers printed before 1922 on the subject of Modular Invariants. The present Appendix is intended to bring Dickson's work up to date. has given a

of

summary

all

Moore* gave a twofold

generalisation of Ferrnat's

Theorem and

showed that the determinant

is

congruent to the product of all distinct non-zero linear forms with n This determinant is none other than the GF[p

coefficients in the

~\.

Lm

which we discussed in 12. Hazlettt discussed the relationship between the theory of modular invariants and the theory of project! ve invariants. She developed the use of symbolical notation and showed the important difference between isobaric and pseudo-isobaric covariants. It is shown that all formal binary invariants admit of symbolical representation. Methods of finding the symbolical representation of an invariant are given. There is an error in this paper which Miss Hazlett J later corrected. universal covariant

Feldstein gave the full system of universal covariants of the w-ary group of transformations whose coefficients are the positive integral viz. residues mod t, where t p 1*

\

pk-i

pk-i

Lm where over

s=l

0, 1,

.

.

?

.

,

.

,

p

w=1 -

1,

j P LjapK-l-^^rftpk-i-i m S m l~\

^,*

'

;

= ,;'

but

1,

'

and where the a and &., s range The above results were the case where k 2 by Turner

may

given in an earlier paper for

...,&-! not

*

all

;

be zero.

||.

Bulletin of the American Mathematical Society, vol. 2, p. 189 (1895-96). t Transactions of the American Mathematical Society, vol. 24, pp. 286-311 (1922). J Ibid. vol. 30, p. 855 (1930). Ibid. vol. 25, pp. 223-238 (1923). Ibid. vol. 24, pp. 129-134 (1922). ||

APPENDIX

79

I

Glenn* gave a full system of the formal covariants of two binary new method of obtaining covariants is quadratics in the GF['2]. " " described which depends upon the appropriate selection of a primary " " no covariants If then the not selection be may quantic. appropriate result, so that this method is of no great value until we have some

A

more

definite method of selecting the primary quantic. Gouwensf extended the results of a paper by Mrs Ballantine t from

m

2 to of

It is

variables.

transformations with

^^PiP^-P^r

sum

a

is

=

proved that every invariant of the group determinant congruent to unity modulo each of which

f invariants,

is

expressible

-

an invariant of the group Hi of transformaA by Pi tions with determinant congruent to unity mod pS. Conversely every such product is an invariant. Williams treated full systems of formal modular protomorphs of as a product of k t

-

binary forms. Elliott formation x = x + ty, y

= 0o

Si

?

1 1

gave the algebraic protomorphs of the trans-

= y,

viz.

$ = 0o 0-2 - 0i 2

2 j

$ =

shown that the seminvariants

It is

form a

full

2

3

- 30

$(=1,

0i

2,

(

-Hf

.J

modp (j= 1,

+

20i

3 5

....

* ...,) and a

system of protomorphs of the binary

such a prime that

2

l-ic

2, ...,/

1

^

modp, where p

is

The theorem

is

1).

-aj"

also given for the case of several binary /-ics.

HazlettU extended the results of a paper by Williams**. Let &m) be any homogeneous polynomial in n variables of order /,

f(i9

>

and

let

fe,

...,

F

be any homogeneous polynomial in the values of f as over the real points of the field. Let % range over

F

#m ) range

F

w) which 1} under transformations of the group r^ are incongruent in the field. If x=-^(pn -~l)ll be an integer, where Ais some fixed positive integer, then any symmetric function of the xth

all

the conjugates of

powers of the FI *

is

,

a formal invariant

of/ under

the group

T^ with

Bulletin of the American Mathematical Society, vol. 30, pp. 131-139 (1924).

f Transactions of the American Mathematical Society, vol. 26, pp. 435-440 (1924).

J American Journal of Mathematics, vol. 45, pp. 286-293 (1923). Transactions of the American Mathematical Society, vol. 28, pp. 183-197 ||

IF

Algebra of Quantics, pp. 212-215. Journal de Mathematique, Ser. 9, vol.

** Ibid. vol. 4, pp. 169-192 (1925).

9, pp.

327-332 (1930).

(1926).

APPENDIX

80

n respect to the GfF[p ]. This result of obtaining formal covariants.

E.

Noethert proved the

is

finiteness of

I

similar to Dickson's*

method

modular covariants by using

the Theory of Fields.

APPENDIX

II

list of papers on the subject of Modular of these are summarised either in Appendix I

In this Appendix we give a Invariants.

The contents

or by Dickson in his History of the Theory of Numbers. the following abbreviations

We

shall use

:

T.A.M.S. = Transactions of JB.

the

American Mathematical

Society.

A.M. 8. = Bulletin of the American Mathematical Society.

A .J.M.

American Journal of Mathematics.

P.L.M.S. = Proceedings of the London Mathematical Q.J.M. Quarterly Journal of Mathematics.

A.M.

Society.

Annals of Mathematics.

= Journal de Mathematique. P. N. A. 8. = Proceedings of the National Academy of Sciences. J.M.

*

Transactions of the American Mathematical Society, vol. 15, pp. 497-503 (1914). (1926), pp. 28-35.

t GWtinger Nachrichten

APPENDIX

II

81

APPENDIX HI

82

APPENDIX We m-ary

III

tabulate here the papers in which the modular covariants of an and /. The numbers are considered for particular values of

/-ic

m

of the papers refer to those of Appendix 1^ 1=1 + 2 denotes that the simultaneous covariants of a linear and a quadratic form are treated.

INDEX Actual sub-rings 53 algebra 52 division 52 algebraic concomitants 4, 12 dependency 60 elements 55 of the first sort 67 expansion 55, 60 algebraically reducible systems 61 annih'ilators 26 Aronhold operators 23 Artin 73 ,

Basis 58 of a module 70 of an ideal 69 of a system 6 rational 62, 63

commutative ring 52 completely isobanc 10, 12 concomitants, algebraic 4, 12 congruent 4, 10, 12 formal 4, 15, 28, 30 ,

non-formal

5,

51

residual 5, 33, 36 congruence, identical 1 residual 1 ,

,

finite

57

of the first sort 67 of the second sort 67

simple 54 transcendental 56, 60 exponent of an element 66 expansion 66 extension 54 ,

,

congruent concomitants conjugate points 5

2,

78

K

finiteness of covariants 76-77 criterion 74

,

4, 10,

12

covariants 4 , universal 15

Degree of an element 55, 65 expansion 57 ,

,

,

,

,

,

Fermat's theorem 1, field 52 finite 52 Galois 1 medial 53 perfect 65 quotient 52 radical 66 sub- 53 super- 53 fields, isomorphic 57 f 65 finite expansion 57 field 52 ideal 69

Capelli 8 characteristic 53 invariants 33, 42 classes 31, 37, 51 cogredient points 5

,

,

Feldstein 78 Ferniat 1, 3

,

modular 5

equivalent systems 60 expansion 54 algebraic 55, 60 degree of 57 exponent of 66, 68

property 6 algebraic elements of the 67 expansions of the 67 polynomials of the 66 formal concomitants 4, 15, 28, 30 invariants 22, 26 full system 6 fundamental system 6 first sort, ,

,

transcendental 62

dependency 56, 60 Dickson 4, 15, 17, 26, 28, 31, 34, 38, 47, 78 Dickson's method for formal covariants 28 theorem 17 division algebra 52 ring 52

Galois field 1

imaginaries 2 generators of linear transformations 8 Glenn 22, 26, 38, 79

Gouwens 79

divisor chains 68, 70

Grace 13 ground forms 4

Elliott 10, 79 entire element 72

syzygies 36 m the order of 3 group r groups of transformations 3

entirely closed fields 72 equivalent rings 57

Hazlett 4, 12, 29, 78

,

IHBEX residual invariants 33 invariants, cubic 51 linear 46 ,

Ideal 68, 69 identically congruent 1 fields

65

imperfect invariant 4 formal 22 of the group 4

quadratic 47 residually congruent 1 ring 52 ,

,

commutative 52 polynomial 53 rings, equivalent 57

irreducible system 61 isobaric 10

,

,

isomorphism 57

,

Krathwohl

Sanderson 4, 7, 37, 39 Sanderson's theorem 39 second sort, expansion of the 67 semin variants 28, 30 simple expansion 54 smallest full system 43

Linear dependency 56 form 22, 45

Medial

field

isomorphic 57

22, 24 ,

53

ring 53

modular Aronhold operator

23,

concomitant 5

.sub-ring 53

polar 23 transvectant 23 module basis 70

'super-field 53

modules 70 Moore 3, 78 Noether, E. 6, 62, 74, 76 Noether's theorem 75 non-formal concomitants

Speiser 1, 3 Steinitz 53, 57, 60, 65 sub-field 53

cogredience 15

super-ring 53 symbolical notation 6, 30 operators 24 system, algebraically reducible 61 basis of 6 systems, equivalent 60 lull 6 fundamental 6 smallest full 6, 43 ,

5,

51

,

,

Order of a Galois field 2 of the group Tm 3 Perfect fields 65 points, cogredient 5 polar, modular 23 polynomial ring 53

polynomials 53 of the first sort 66 pseudo-isobarism 15, 30 pure transcendental expansions 61

Quadratic ground form 47 quotient field 52 .R-entire elements 72 JR-modules 70 JJ-polynomials 53

radical fields 66] rational basis theorem 62 reducible systems 61 residual concomitants 5, 33, 36, 45, 47, 51 covariants 36, 51

,

syzygies 35

Transcendental degree 62 element 56 expansion 56, 60 pure, 61 ,

transformations, generators of 8 groups of 3 transvectants, modular 23 Turner 78 ,

Universal covariants

4,

15

Van

der Waerden 68, 72 van der Waerden' s theorem 73

Weight 10 Weitzenbock

13, 14, 35

Williams 79

Young 13

PBINTED BY WALTER LEWIS, M.A., AT THE UNIVERSITY PRESS, CAMBRIDGE