Elements of Abstract Algebra

236 Pages·2007·32.52 MB·English

Elements of
Abstract Algebra
Allan Clark
Purdue University
Dover Publications, Inc.
New York
For my parents
I was just going to say, when I was interrupted, that one of the
many ways of classifying minds is under the heads of arith
metical and algebraical intellects. All economical and practical
wisdom is an extension of the following arithmetical formula:
2 + 2 = 4. Every philosophical proposition has the more
general character of the expression a + b = c. We are mere
operatives, empirics, and egotists until we learn to think in letters
instead of figures.
OLIVER WENDELL HOLMES
The Autocrat of the Breakfast Table
Copyright © 1971, 1984 by Allan Clark.
All rights reserved under Pan American and International Copyright
Conventions.
Published in Canada by General Publishing Company, Ltd., 30
Lesmill Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd.,
10 Orange Street, London WC2H 7EG.
This Dover edition, first published in 1984, is an unabridged and
corrected republication of the work first published by Wadsworth Pub
lishing Company, Belmont, California, in 1971.
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N. Y. 11501
Library of Congress Cataloging in Publication Data
Clark, Allan, 1935-
Elements of abstract algebra.
"Corrected republication"-Verso t.p.
Originally published: Belmont, Calif. : Wadsworth, © 1971.
Bibliography: p.
Includes index.
I. Algebra, Abstract. l. Title.
[QAI62.C57 1984] 512'.02 84-6118
ISBN 0-486-64725-0
Foreword
Modern or" abstract" algebra is widely recognized as an essential element
of mathematical education. Moreover, it is generally agreed that the axioma
tic method provides the most elegant and efficient technique for its study.
One must continually bear in mind, however, that the axiomatic method is an
organizing principle and not the substance of the subject. A survey ofalgebraic
structures is liable to promote the misconception that mathematics is the
study of axiom systems of arbitrary design. It'seems to me far more interesting
and profitable in an introductory study of modern algebra to carry a few topics
to a significant depth. Furthermore I believe that the selection of topics should
be firmly based on the historical development of the subject.
This book deals with only three areas of abstract algebra: group theory,
Galois theory, and classical ideal theory. In each case there is more depth
and detail than is customary for a work of this type. Groups were the first
algebraic structure characterized axiomatically. Furthermore the theory of
groups is connected historically and mathematically to the Galois theory of
equations, which is one of the roots of modern algebra. Galois theory itself
gives complete answers to classical problems of geometric constructibility and
solvability of equations in radicals. Classical ideal theory, which arose from
the problems of unique factorization posed by Fermat's last theorem, is a
natural sequel to Galois theory and gives substance to the study of rings. All
three topics converge in the fundamental theorem of algebraic number theory
for Galois extensions of the rational field, the final result of the book.
Emil Artin wrote: We all beliel'e that mathematics is an art. The author 0/ a
book, the lecturer in a classroom tries to convey the structural beauty 0/ mathe
matics to his readers, to his listeners. In this allempt he must always(ail. Mathe-
v
VI Foreword
matics is logical to be sure; each conclusion is drawn from previously derived
statements. Yet the whole of it, the real piece of art, is not linear; worse than
that its perception should be instantaneous. We all have experienced on some
rare occasions the feeling of elation in realizing that we have enabled our list
eners to see at a moment's glance the whole architecture and all its ramifications.
How can this be achieved? Clinging stubbornly to the logical sequence inhibits
visualization of the whole, and yet this logical structure must predominate or
chaos would result. t
A text must cling stubbornly to the logical sequence of the subject. A lec
turer may be peripatetic, frequently with engaging results, but an author must
tread a straight and narrow path. However, though written sequentially, this
book need not be read that way. The material is broken into short articles,
numbered consecutively throughout. These can be omitted, modified, post
poned until needed, or given for outside reading. Most articles have exercises,
a very few of which are used later in proofs. What can be covered in an or
dinary course and for what students the text is suitable are questions left to
the instructor, who is the best judge of local conditions. It is helpful, but
certainly not essential, for the reader to know a little linear algebra for the
later chapters- in particular Cramer's rule. (Vector spaces, bases, and dimen
sion are presented in articles 90-95.)
Finally, 1 must gratefully acknowledge the assistance of Mrs. Theodore
Weller and Miss Elizabeth Reynolds, who typed the manuscript, and the
help of Messrs. George Blundall and John Ewing, who gave their time and
patience to proofing it.
Providence, Rhode Island
January 1, 1970
t Bulletin of the American Mathematical Society, ) 953, p. 474. Reprinted by permission
of the publisher.
Contents
Foreword v
Introduction ix
One Set Theory
1-9 The notation and terminology of set theory 2
10-16 Mappings 5
17-19 Equivalence relations 9
20-25 Properties of natural numbers 1 I
Two Group Theory
26-29 Definition of group structure I7
30-34 Examples of group structure 22
35-44 Subgroups and cosets 24
45-52 Conjugacy, normal subgroups, and quotient groups 32
53-59 The Sylow theorems 39
60-70 Group homomorphism and isomorphism 45
71-75 Normal and composition series 53
76-86 The Symmetric groups 57
vii
VIII Contents
Three Field Theory
87-89 Definition and examples off ield structure 67
90-95 Vector spaces, bases, and dimension 69
96-97 Extension fields 73
98-107 Polynomials 75
108-114 Algebraic extensions 88
115-121 Constructions with straightedge and compass 95
Four Galois Theory
122-126 Automorphisms 104
127-138 Galois extensions 108
139-149 Solvability of equations by radicals 130
Five Ring Theory
150-156 Definition and examples of ring structure 145
157-168 Ideals 151
169-175 Unique factorization 160
Six Classical Ideal Theory
176-179 Fields off ractions 174
180-187 Dedekind domains 179
188-191 Integral extensions 186
191-198 Algebraic integers 188
Bibliography 197
Index 199
Introduction
Classical algebra was the art of resolving equations. Modern algebra, the
subject of this book, appears to be a different science entirely, hardly con
cerned with equations at all. Yet the study of abstract structure which charac
terizes modern algebra developed quite naturally out of the systematic
investigation of equations of higher degree. What is more, the modern
abstraction is needed to bring the classical theory of equations to a final
perfect form.
The main part of this text presents the elements of abstract algebra in a
concise, systematic, and deductive framework. Here we shall trace in a
leisurely, historical, and heuristic fashion the genesis of modern algebra from
its classical origins.
The word algebra comes from an Arabic word meaning " reduction" or
"restoration." It first appeared in the title of a book by Muhammad ibn
Musa al-Khwarizmi about the year 825 A.D. The renown of this work, which
gave complete rules for solving quadratic equations, led to use of the word
algebra for the whole science of equations. Even the author's name lives on
in the word algorithm (a rule for reckoning) derived from it. Up to this point
the theory of equations had been a collection of isolated cases and special
methods. The work ofal-Khwarizmi was the first attempt to give it form and
unity.
The next major advance came in 1545 with the publication of Artis Magnae
IX
x Introduction
sive de Regulis Algebraicis by Hieronymo Cardano (1501-1576). Cardano's
book, usually called the Ars Magna, or" The Grand Art," gave the complete
solution of equations of the third and fourth degree. Exactly how much
credit for these discoveries is due to Cardano himself we cannot be certain.
The solution of the quartic is due to Ludovico Ferrari (1522-1565), Cardano's
student, and the solution of the cubic was based in part upon earlier work of
Scipione del Ferro (1465?-1526). The claim of Niccolo Fontana (1500?-
1557), better known as Tartaglia (" the stammerer "), that he gave Cardano
the cubic under a pledge of secrecy, further complicates the issue. The bitter
feud between Cardano and Tartaglia obscured the true primacy of del Ferro.
A solution of the cubic equation leading to Cardano's formula is quite
simple to give and motivates what follows. The method we shall use is due
to Hudde, about 1650. Before we start, however, it is necessary to recall that
every complex number has precisely three cube roots. For example, the com
plex number! = 1 +Oi has the three cube roots, I (itself), w = -t + t V-3,
and w2 = -t - t '\[=3. In general, if z is anyone of the cube roots of a com
plex number w, then the other two are wz and w2z.
For simplicity we shall consider only a special form of the cubic equation,
x3 + qx - r = O. (1)
(However, the general cubic equation may always be reduced to one of this
form without difficulty.) First we substitute u + v for x to obtain a new
equation,
(2)
which we rewrite as
u3 + v3 + (3uv + q)(u + v) - r = O. (3)
Since we have substituted two variables, u and v, in place of the one variable
x, we are free to require that 3uv + q = 0, or in other words, that v = -q/3u.
We use this to eliminate v from (3), and after simplification we obtain,
(4)
This last equation is called the resolvent equation of the cubic (I). We may
view it as a quadratic equation in u3 and solve it by the usual method to
obtain
(5)
Of course a complete solution of the two equations embodied in (5) gives six
values of u- three cube roots for each choice of sign. These six values of u are
Introduction Xl
the roots of the sixth-degree resolvent (4). We observe however that if u is a
cube root of (r/2) + J(r2/4) + (q3/27), then v = -q/3u is a cube root of
(r/2) - J(r2/4) + (q3/27). Consequently the six roots of (4) may be con
veniently designated as u, WU, w2u and v, wv, w2v, where uv = -q/3. Thus the
three roots of the original equation are
IXI = U + v, (6)
where
J?7
-q
u3 = 2r + "4 + 27 and v=3-u .
In other words, the roots of the original cubic equation (1) are given by the
formula of Cardano,
1X=3/~+Jr2 q3 3/~_Jr2 q3
V V
2 4 + 27 + 2 4 + 27'
in which the cube roots are varied so that their product is always - q/3.
For our purposes we do not need to understand fully this complete solution
of the cubic equation-only the general pattern is of interest here. The im
portant fact is that the roots of the cubic equation can be expressed in terms
of the roots of a resolvent equation which we know how to solve. The same
fact is true of the general equation of the fourth degree.
For a long time mathematicians tried to find a solution of the general
quintic, or fifth-degree, equation without success. No method was found to
carry them beyond the writings of Cardano on the cubic and quartic. Con
sequently they turned their attention to other aspects ()f the theory of equa
tions, proving theorems about the distribution of roots and finding methods
of approximating roots. In short, the theory of equations became analytic.
One result of this approach was the discovery of the fundamental theorem
of algebra by 0' Alembert in 1746. The fundamental theorem states that every
algebraic equation of degree n has n roots. It implies, for example, that the
equation X' - 1 = 0 has n roots-the so-called nth roots of unity-from which
it follows that every complex number has precisely n nth roots. D'Alembert's
proof of the fundamental theorem was incorrect (Gauss gave the first correct
proof in 1799) but this was not recognized for many years, during which the
theorem was popularly known as " D' Alembert's theorem."
D'Alembert's discovery made it clear that the question confronting alge
braists was not the existence of solutions of the general quintic equation, but
whether or not the roots of such an equation could be expressed in terms of
its coefficients by means of formulas like those of Cardano, involving only
the extraction of roots and the rational operations of addition, subtraction,
multiplication, and division.
Xli Introduction
In a new attempt to resolve this question Joseph Louis Lagrange (1736-
1813) undertook a complete restudy of all the known methods of solving
cubic and quartic equations, the results of which he published in 1770 under
the title Refiexions sur fa resolution algebrique des equations. Lagrange
observed that the roots of the resolvent equation of the cubic (4) can be ex
pressed in terms of the roots exl, ex2, ex3 of the original equation (1) in a com
pletely symmetric fashion. Specifically,
v = -t(exl + wex2 + W2ex3), u = -t(exl + wex3 + W2ex2),
wv = -t(ex3 + wexl + w2ex2), wu = -t(ex2 + wex1 + W2ex3), (7)
All these expressions may be obtained from anyone of them by permuting
the occurrences of exl, ex2, ex3 in all six possible ways.
Lagrange's observation was important for several reasons. We obtained
the resolvent of the cubic by making the substitution x = u + v. Although
this works quite nicely, there is no particular rhyme nor reason to it-it is
definitely ad hoc. However Lagrange's observation shows how we might have
constructed the re,solvent on general principles and suggests a method for
constructing resolvents of equations of higher degrees. Furthermore it shows
that the original equation is solvable in radicals if and only if the resolvent
equation is.
To be explicit let us consider a quartic equation,
(8)
and suppose that the roots are the unknown complex numbers exl, ex2, ex3, ex4.
Without giving all the details we shall indicate how to construct the resolvent
equation. First we recall that the fourth roots of unity are the complex
=J-=1
numbers I, i, i2, i3, where i and i2 = -I, i3 = - i. Then the roots of
the resolvent are the twenty-four complex numbers
(9)
where the indices i, j, k, f are the numbers I, 2, 3, 4 arranged in some order.
Therefore the resolvent equation is the product of the twenty-four distinct
factors (x - Uijkl). That is, we may write the resolvent equation in the form
¢(x) = TI (x - Uijkl) = o. (10)
ijkl
Thus the resolvent of the quartic has degree 24, and it would seem hopeless
to solve. It turns out, however, that every exponent of x in ¢(x) is divisible by