Modern Algebraic Theories

Table of contents :
Front Cover
Title Page (Page i)
Copyright (Page ii)
Table of Contents (Page vii)
Section 1 (Page 1)
Section 2 (Page 24)
Section 3 (Page 39)
Section 4 (Page 64)
Section 5 (Page 89)
Section 6 (Page 112)
Section 7 (Page 135)
Section 8 (Page 150)
Section 9 (Page 159)
Section 10 (Page 178)
Section 11 (Page 204)
Section 12 (Page 210)
Section 13 (Page 220)
Section 14 (Page 251)
Section 15 (Page 271)
Index (Page 272)

Citation preview

II1MI|IJ1I|IJII:IMI!UMII1UM

TIIK tJIFT OK Prof. Alexander Ziwet

=====

11

J '\

GNUlNc. T\V,■ LIBRA;;

-

JSS' PJ5SJ

MODERN

ALGEBRAIC THEORIES LEONARD E. DICKSON, Ph.D. CORRESPONDANT

DB L INSTITUT DB FRANCE

PROFESSOR OF MATHEMATICS

IN THE UNIVERSITY OF CHICAGO

BENJ. H. SANBORN CHICAGO

NEW YORK

& CO. BOSTON

COPYIIGHT,

1936

Bv BENJ. H. SANBORN & CO.

BEQUEST OF PROF. ALEXANDER ZIWET.

PREFACE The rapidly increasing number of students beginning graduate work are handicapped by the lack of books in English which provide readable introductions to important parts of mathematics. Nor is the difficulty met adequately by the slow method of lecture courses.

Purpose of this book

This book is based on the author's lectures of recent

It

presupposes calculus and elementary theory of algebraic equations. Its aim is to provide a simple introduction to the essentials of each of the branches of modern algebra, years.

a

V

is

is

it

is

(j)

with the exception of the advanced part treated in the author's Algebras and Their Arithmetics. The book develops the theories which center around matrices, invariants, and groups, which are among the most important concepts in mathematics. The book provides adequate introductory courses It is a text for several courses hi higher algebra, (ii) the Galois theory of algebraic equations, (iii) finite linear groups, including Klein's "icosahedron" and theory of equations of the fifth degree, and (iv) algebraic invariants. The subject known in America as higher algebra Higher algebra treated fully in Chapters III-VI; includes matrices, linear transformations, elementary divisors and invariant factors, and quadratic, bilinear, and Hermitian forms, whether taken singly new or in pairs. While the results are classic, the presentation Due attention given to ques and particularly elementary. tions of rationality, which are too often ignored. The unified treatment of Hermitian and quadratic forms requires but little more space than would be needed for quadratic forms alone. Elementary divisors and invariant factors are introduced in in simple, natural way in connection with the classic Chapter and a new rational canonical form of linear transformations; this iii

iv treatment

PREFACE is not only more elementary than the usual one, but

develops these topics in close connection with their most frequent applications. It is then a simple matter to deduce in Chapter VI the theory of the equivalence of pairs of bilinear, quadratic, or Hermitian forms. We thereby avoid the extraneous topic of matrices whose elements are polynomials "elementary transformations" of them.

in a variable and the

Is every equation solvable by radicals? This question js one Gf absorbing interest in the history of mathe matics. It was finally answered in the negative by means of groups of substitutions or permutations of letters. The usual presentation of group theory makes the subject quite abstruse. This impression is avoided in the exposition in Chapters VII-XI. Algebraic

equations

Substitutions are introduced in a very deliberate and natural way in connection with the solution of cubic and quartic equa tions. The reader will therefore appreciate from the start some of the reasons why substitutions and groups are employed. Fortunately we are able to alternate theory and application in the further exposition of groups. The theory gives very simple answers to the following questions: Can every angle be trisected with ruler and compasses? What regular polygons can be con structed by elementary geometry? Klein's book on the icosahedron and equations of Icosahedron, linear groups the fifth degree is a classic, but causes real diffi culties to beginners on account of the inclusion of ideas from many branches of higher mathematics. Chapter XIII gives a simple exposition of the essentials of this interesting theory, which is a prerequisite to the subjects of elliptic modular func tions and automorphic functions. The preliminary Chapter XII discusses the removal of several terms from any equation by means of a rational (Tschimhaus) transformation, and the reduc tion of the general equation of the fifth degree to Brioschi's normal form, which is well adapted to solution by elliptic functions. The final Chapter XIV is a sequel to Klein's theory. It establishes remarkable results on the representation of a given group as a

PREFACE

v

linear group, and gives an introduction to Frobenius's theory of group characters. The latter is an effective tool for finite groups and has led to important new results.

I and II

provide an easy introduction to the important subject of invariants. Hessians and Jacobians are shown to be covariants and applied to the determina

Algebraic invariants

Chapters

tion of canonical forms of binary cubic and quartic forms, as well as to the solution of cubic and quartic equations. Every seminvariant is proved to be the leading coefficient of one and only one covariant. It is shown that all covariants of any system of binary forms are expressible in terms of a finite number

of the covariants.

Such a fundamental

system of covariants is actually found for one form of each of the orders 1, 2, 3, 4. Valu able supplementary work on invariants is provided by Chapter

XIII,

which presupposes the concept of groups of substitutions explained in the elementary Chapter VII. There are numerous sets of simple problems, and a few historical notes. On pages 38, 133, 176, 203, and 249 there are lists of topics for further reading, with references to writings in English. These topics are suitable for assignment to students for full reports at the end of the particular course. University of Chicago March 16, 1926

L. E. Dickson

CONTENTS CHAPTER

I.

II.

III.

Introduction to Algebraic Invariants

Linear transformations. Hessians. Invariants and covariants. Jacobians. Discriminants. Canonical forms of binary cubic and quartic forms. Solution of cubic and quartic equations. Homogeneity. Weights. Seminvariants. Fundamental system of covariants of a binary p-ic for p 4 not solvable by radicals. Solvable quintics.

XI.

XII.

112

135

150

159

178

Constructions with Ruler and Compasses Analytic criterion for constructibility. Trisection of an angle. Regular polygons.

204

Reduction of Equations to Normal Forms Tschirnhaus transformations. Principal equations. The BringJerrard normal form. Brioschi's normal form of quintic equa tions.

210

CONTENTS CHAPTER

XIII.

XIV.

. Groups of the Regular Solids; Quintic Equations Linear fractional transformation corresponding to a rotation. Tetrahedral, octahedral, and icosahedral groups; their invari ants and form problems. Principal quintic resolvent of the icosahedral equation; its identification with any principal quintic. General quintic. Transformation of Brioschi's re solvent into the principal resolvent. Galois group of the icosahedral equation. Further results stated.

Representations of a Finite Group as a Linear Group; Group Characters Reducible linear groups. Representations. Irreducible and re ducible group matrices. Regular group matrix. Group char acters. Applications to group matrices. Alternating group on five letters. Computation of group characters.

ix PAGE „

220

251

Subject Index

271

Author Index

275

!

i

J

MODERN ALGEBRAIC THEORIES Chapteb INTRODUCTION

I

TO ALGEBRAIC

INVARIANTS

Invariants and covariants play an important r61e in the various parts of modern algebra as well as in geometry. The elementary theory presented in this chapter will meet the ordinary needs in other parts of mathematics. It covers rather fully the subject of invariants and covariants of a homogeneous polynomial in x and y of degree < 5, with application to the solution of cubic and quartic equations. When the degree exceeds 4, most of the cova riants are too long to be of real use, unless their symbolic repre sentation is employed.

Linear transformations. When two pairs of variables x, y and £, V are connected by relations of the form 1.

x = a£ + br),

y =

c^

+

D

dV,

a

=

c

b

d

*0,

these relations define a linear transformation T of determinant D. Consider another linear transformation U defined by Z

=

eX+fY,

r,

=

gX + hY,

The equations obtained by eliminating x =

kX + lY,

A = £

and

V

e

f

g h

5*0.

are of the

form

y = mX + nY,

and define a linear transformation P which is called the product of T and U, taken in that order, and is denoted by TU. The values of the coefficients are

1

ALGEBRAIC INVARIANTS

2

k = ae

+

m = ce

I = af + bh,

bg,

+

[Ch.

I

n = cf + dh.

dg,

The determinant of P is found to be equal to DA, and hence is not zero. If we solve the equations of T and in the result replace x by X and y by Y, we get

f

=

- D-1 bY,

D-1 dX

v =

- D-1 cX + D-1 aY.

These equations define a transformation called the inverse of T and denoted by T~K Since the variables of T-1 are the same as those of U, the product TT-1 is found by eliminating f and n and hence is x

—

X,

y = Y.

The latter is called the identity is readily verified that also T-1 T is the identity

transformation. It transformation. We shall next prove the associative law

TU V .

which allows us to write

X

TUV

=

T

.

UV,

for either product.

= pu + gv,

Y

of V. The product

—

ru +

TU V

Let

sv

by elim inating first £, ij and then X, Y between the equations of T, U, V, while the product T . UV is obtained by eliminating first X, Y be the equations

and then

.

In

each case we must evidently obtain the same equations expressing x and y as linear functions of u and v. 2.

$, r\

between those equations.

is found

Forms and their classification. ax3

+

bx1

A polynomial like

y + cxyz + dxz2,

every term of which is of the same total order, (here 3) in the vari ables x, y, z, is called homogeneous in x, y, z. A homogeneous polynomial is called a form. According as the number of variables

HESSIANS

§3] is 1, 2, 3, 4, quaternary, .

or q, the form is called unary, binary, ternary, or q-ary, respectively. According as the order of

. . . , . . ,

the form is 1, 2, 3, 4,

quartic,

3

p, it is called linear, quadratic, cubic,

. . . ,

respectively. For example, the polynomial displayed above is a ternary cubic form, while ax2 + bxy + cy2 is a binary quadratic form. . . . ,

p^ic,

Hessians. The Hessian (named after Otto Hesse) of a func tion /(x, y) of two variables is the determinant 3.

37 dx2

(1)

d2f

dydx

d2f

dxdy

*f dy2

For example, the

whose elements are second partial derivatives. Hessian of = ax2 + 2bxy + cy2 is 4 (ac — b2).

/

The Hessian h of f(xi, . . . , xt) is the determinant of order which the elements of the ith row are (2)

Let (3)

d2f

d2f

d2f

dXi dxi

dXi dxt

dXi dxq

/ become F(yi, xi = cayi+

. . . ,

q

in

yq) under the linear transformation

\-city,

(*

«

1, . . . ,

q),

whose determinant is cll

clq

cql

is a determinant of order q whose element in Ihe«product the ith row and jth column is the sum of the products of the

ALGEBRAIC INVARIANTS

4

[Ch.

I

elements (2) of the ith row of h by the corresponding elements cij, ct cqj of the jth column of A and hence is equal to the

partial derivative with respect to x < of df cij + J-

,

+

^cj dF

dxi

dx

dxi dyt

dxq

dyj

df-

(4)

df — cij + df

,

dyt

S

d

r

d

dF

dF

dxq

dyj

dyr dyj

the partial derivative of Xi in (3) with respect to yr

by

the

Hessian

of

fof

by

of

A,

determinant any linear transformation A2. the Hessian equal to the product

applying is

function obtained from

f

1.

the

Hessian of F.

i-i,

F

dyr dyj r,

of

d2F

=

If

A

A2

Theorem

+

dxi dyf

is

Hence

dF

—

F

since cir

is

clr

c

is

h

A,

A

by interchang Let A' denote the determinant obtained from ing its rows and columns, whence A' = A. In the product A' • the element in the rth row and jth column therefore

Definition of invariants and covariants. Let transforma of determinant linearly in terms of tion expressing x and = ax2 + 2bxy + cy2 by A£2 and ij, replace 2B%rj + C?j2. We saw in that the Hessian of the product of the discriminant of by ac — By Theorem +

y

1, is

/

AC

-

B2 = A2(oc

-

/.

We therefore call the discriminant ac — of We shall generalize this definition.

b2). b2

4.

/

b2

§3

£

/

T

A,

a

4.

an invariant of ipdex^2

§

DEFINITION OF COVARIANTS

4]

Consider the general binary form of order p,

f= Let

a

transformation T replace

F A polynomial

= A0

/ (oo,

£

/ by

f-1

+ Ai

»

v

+

h ap

.••

y.

+ ApVp.

is called an invariant of indcxj. of of determinant A^O, for every transformation ap )

!T

. . . ,

Ap) =

A*

/(ao,

. .

.

.

,

.

,

I(A0,

.

if,

aox* + al x"-1 y +

5

/

ap),

V)

Ap;

£,

. .

,

K(Aa,

.

/

T

,

identically in Oo, . . . ap, after the A's have been replaced by their values in terms of the a's. Covariants K are defined similarly. If, for every transformation of determinant A^O, a polynomial K in the coefficients and variables of has the property that = A'K(a0,

...

,ap;x, y),

is

v

£

/.

2

is

.

,

2

is

A

is

A

is

,

/i,

.

is

,

.

/

a

a

I

F

a

/

a

is

a

/

1,

is

/.

I

a

T, y

£,

,

identically in a0, . . . aP, i7, after the A's have been replaced have been by their values in terms of the a's, and after x and and from then K replaced by their values in terms of called covariant of index of By Theorem the Hessian of covariant of index of = Note that itself covariant of index zero of since A0/. of By covariant of index system of forms /i, . ,/* meant function of their coefficients and variables xi, . . . xq whose product by A' equal to the same function of the corre sponding coefficients and variables yi, . . yq in the forms ...,/* by applying the general Fi, . . . Fk derived from linear transformation (3) whose determinant not zero. An below. given by Theorem example covariant which does not involve any of the variables an invariant.

ALGEBRAIC INVARIANTS

6

[Ch.

I

Jacobians. The functional determinant or Jacobian (named ...,x,),... after C. G. J. Jacobi) of q functions . . . , xq) with respect to the variables Zi, . . . , xq is defined to be 5.

the determinant

#L

dxy

dXi

dxi

dXi

...

#L

dxq

dxq

Theorem 2. The Jacobian of fi, . . . unity of the system of forms fi, . . . ,fq.

,

/,

is a covariant of index

Under a transformation (3) of determinant A, let /< become PikUi, • • • » Vq)- Then (4) holds if we put the subscript i on each and F. Hence the Jacobian of Fi, . . . , Fq with respect to the variables yi, . . . , yt is equal to

/

dfi — cu

-\

cu

I

-

+

dfi —

-f

^'cl...

cqi

• . •

dfi — ciq

cl

H

i

h

i

a/i — c49

a/