Elements of Abstract Algebra

221 Pages·1971·24.81 MB·English

by  Allan Clark

ELEMEONFT S
ABSTRACT
ALGEBRA
ALLCALNA RK
Elemeonft s
AbstraAcltg ebra
Allan Clark
Purdue University
D vePurb lait iIonn.
N w � rk
Fomry p arents
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 propositioJJ. 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.
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.
1. Algebra, Abstract. I. Title.
[QA162.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 of algebraic
structures is liable to promote the misconception that mathematics is the
study of axiom systems of arbitrary design. I rseems 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 helicre that mathematics is an art. The author of a
book, the lecturer in a classroom tries to conrey the strnctural beauty of mathe­
matics to !,is readers, to his listeners. In this alle mpt he must a/ways.fail. Mathe-
V
YI Foreword
ma tics 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
t
chaos would result.
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, I 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 l, 1970
t Bulletin of the American Mathematical Society, 1953, p. 474. Reprinted by permission
of the publisher.
Contents
Foreword v
Introduction ix
One Set Theory
l-9T he notation and terminology of set theory 2
10-16 Mappings 5
17-19 Equivalence relations 9
20-25 Properties of natural numbers 11
Two Group Theory
26-29 Definition of group structure 17
30-34 Example of group structure 22
35--44 ubgroups and coset 24
4 52 onjugacy, normal subgroup , and quotient groups 32
5 59 The ylow theorem 39
60-70 roup homomorphi m and i omorphi m 45
71-75 ormal and ompo ition eri 53
7 86 h mm tric group 57
II
VIII Contents
Three Field Theory
87-89 Definition and examples offi eld 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 Sohiability 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 of fractions 174
180-187 Dedekind domains 179
188-191 Integral extensions 186
191-198 Algebraic integers 188
Bibliography 197
Index 199
Introduction
Classailcgaelwb arsta h aer otf r esolevqiunagt iMoondse.rna lgebtrhae,
subjoefct th ibso oka,p peatrobs ea d iffersecnite ennctei rhealrydc,lo yn ­
cernweidte hq uataitoa nlYsle .tt h set uodfya bstrsatcrtu cwthuirccehh arac­
terimzoedse ranl gebdreav eloqpueidtn ea turoaultlo yft hes ystematic
investiogfae tqiuoant ioofhn isg hdeerg reWeh.at i sm oret,h em odern
abstraicsnt ieoendt eodb rintgh cel asstihceaoolrf ye quatitooan fisn al
perffeocrtm .
Them aipna rotft hitse xptr esetnhteesl emeonfat bss traalcgte ibnra a
concissyes,t emaantdid ce,d uctfirvaem ewoHrekr.we e s hatlrla icnea
leisuhriesltyo,ar nidch aelu,r ifassthiicto hnge e neosfmi osd erna lgefbrroam
itcsl assoirciagli ns.
Thew oradlg ebra comefsr oamn A rabwiocr mde ani·n·gr educotri on"
"'restorIatfit risaotpn p.e'a'ir net dh tei tolfae b oobky M uhammaidb n
Musaal -Khwaarbiozutmthiy e e a8r2 A5. nT.h er enoowfnt hiwso rkw.h ich
gavceo mplreutlefoe rss olvqiunagd reaqtuiact iloentdso u, s oef t hweo rd
algefobrrt ah weh olsec ieonfce eq uatiEovnestn.h a eu thonra'msle i voens
int hweo radlg orithm (ar ulfeor re ckondienrgi)fv reodim tU .p t ot hip�o int
thteh eoorfey q uathiaodnb se ean c ollecotfi isoonl actaesaden sd s pecial
methoTdhsew. o rokf a l-Khwawraistz hmfeii rasttt etmopg ti vietfo rma nd
unity.
Then exmta joard vanccaemi en 1 54w5i tthh peu blicoafAt ritiso Mna gnac
IX
x Introduction
sive de Regulis A lgebraicis by Hi eronymo 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 C' 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 H udde, 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 numberl = 1 +O i has the three cube roots, 1 (itself), w = --½ +-½ -v-=1,
--½--½ Y-3.
and w2 = In general, if z is any one 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 = 0. (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,
(u3 + 3u2v + 3uv2 + v3) + q(u + v) - r = 0, (2)
which we rewrite as
u3 + v3 + (3uv + q)(u + n) - r = 0. (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 5implification we obtain,
3
u6 - ru3 - 9.__ = 0. (4)
27
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
2 3
u3 =-+ � +-. (5)
-
r2 Jr4 q27
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 xi
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
(6)
where
r r2 3
u 3 = 2 + 4 + q and V=--q.
J 27 3u
In other words, the roots of the original cubic equation (1) are given by the
formula of Cardano,
a=J;+ J': +;;+J;-J': +;�.
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
rs
of the cubic equation-only the general pattern 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 of 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 D'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 - 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'A lembert'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.