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.