Algebraic theories

308 Pages·1959·10.066 MB·English

by  Dickson, Leonard Eugene

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ALGEBRAIC THEORIES

ALGEBRAIC THEORIES
(Formerly Titled: Modern Algebraic Theories)
By
LEONARD E. DICKSON
DOVER PUBLICATIONS, INC.
NEW YORK
Copyright, 1926
By BENJ. H. SANBORN & CO.
This new Dover edition first published in 1959
is an unabridged and unaltered republication
of the First Edition which was formerly
published under the title Modern Algebraic
Theories.
Manufactured in the United States of America.
Dover Publications, Inc.
180 Varick Street
New York 14, N. Y.
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 is based on the author’s lectures of recent
this book years. 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,
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.
It is a text for The book provides adequate introductory courses
several courses ¡n 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.
Higher The subject known in America as higher algebra is
algebra treated fully in Chapters III-VI; it includes matrices,
linear transformations, elementary divisors and invariant factors,
and quadratic, bilinear, and Hermitian forms, whether taken singly
or in pairs. While the results are classic, the presentation is new
and particularly elementary. Due attention is given to ques­
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
Chapter V in a simple, natural way in connection with the classic
and a new rational canonical form of linear transformations; this
iii
IV PREFACE
treatment 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 in a variable and the
“elementary transformations” of them.
Algebraic Is every equation solvable by radicals? This question
equations js one 0f 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.
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?
Icosahedron, Klein’s book on the icosahedron and equations of
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.
Algebraic Chapters I and II provide an easy introduction to the
invariants important subject of invariants. Hessians and Jaco-
bians are shown to be covariants and applied to the determina­
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 fists 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.
L. E. Dickson
University of Chicago
March 16, 1926