Basic Algebra I: Second Edition

514 Pages·2016·14.06 MB·English

by  Nathan Jacobson

Basic ~ l ~ e b r a / l
Second Edition
N A T H A N ~ O B S O N
VALE UNIVERSIW
W. H. FREEMAN AND COMPANY
New York
Library of Congress Cstalogiog in Publiation Data
Jacobson, Nathan, 1910-
Basic algebra.
Includes index.
1. Algebra. I. Title.
QA154.2J32 1985 512.9 8425836
ISBN 0-7167-1480-9 (v. I)
Copyright @ 1985 by W.H . Freeman and Company
No part 01 this book may be reproduced
by any mechanical, photographic, or electronic process,
or in the lorn of a phonographic recording.
nor may it be stored in a retrieval system, transmitted,
or otherwise copied lor public or private use,
without written permission hom the publisher.
Printed in the United States of America

Contents
Preface xi
Preface to the First Edition xiii
INTRODUCTION: CONCEPTS FROM SET THEORY.
THE INTEGERS 1
0.1 The power set of a set 3
0.2 The Cartesian product set. Maps 4
0.3 Equivalence relations. Factoring a map through an
equivalence relation 10
0.4 The natural numbers 15
0.5 The number system E of integers 19
0.6 Some basic arithmetic facts about E 22
0.7 A word on cardinal numbers 24
1 MONOIDS AND GROUPS 26
1.1 Monoids of transformations and abstract monoids 28
1.2 Groups of transformations and abstract groups 31
1.3 Isomorphism. Cayley's theorem 36
1.4 Generalized associativity. Commutativity 39
1.5 Submonoids and subgroups generated by a subset. Cyclic groups 42
1.6 Cycle decomposition of permutations 48
1.7 Orbits. Cosets of a subgroup 51
1.8 Congruences. Quotient monoids and groups 54
1.9 Homomorphisms 58
'
1.10 Subgroups of a homomorphic image.
Two basic isomorphism tbeorems 61
1.11 Free objects. Gent-ators and relations 67
1.12 Groups acting on sets 71
1.13 Sylow's theorems 79
2 RINGS 85
Definition and elementary properties 86
Types of rings 90
Matrix rings 92
Quaternions 98
Ideals, quotient rings 101
Ideals and quotient rings for Z 103
Homomorphisms of rings. Basic theorems 106
Anti-isomorphisms 11 1
Field of fractions of a commutative domain 115
Polynomial rings 119
Some properties of polynomial rings and applications 127
Polynomial functions 134
Symmetric polynomials 138
2.14 ~actondm inor& and nngs 140
2 15 Pnnnoal deal domano and Euclidean domans 147
2.16 Polynomial extensions of factorial domains 151
2.17 "Rngs" (rings without unit) 155
3 MODULES OVER A PRINCIPAL IDEAL DOMAIN 157
3.1 Ring of endomorphisms of an abelian group 158
3.2 Left and right modules 163
3.3 Fundamental concepts and results 166
3.4 Free modules and matrices 170
3.5 Direct sums of modules 175
3.6 Finitely generated modules over a p.i.d. Preliminary results 179
3.7 Equivalence of matrices with entries in a p.i.d. 181
3.8 Structure theorem for finitely generated modules over a p.i.d. 187
3.9 Torsion modules, primary components, invariance theorem 189
3.10 Applications to abelian groupsand to linear transformations 194
3.1 1 The ring of endomorphisms of a finitely generated module
over a p.i.d. 204
4 GALOIS THEORY OF EQUATIONS 210
4.1 Preliminary results, some 014 som new 213
4.2 Construction with strwt-edge and compass 216
4.3 Splitting field of a polynomial 224
4.4 Multiple rwts 229
dairing
4.5 The Galois group. The fundamental Galois 234
4.6 Some mults on hite groups 244
4.7 Galois' criterion for solvability by radicals 251
4.8 The Galois group as permutdion group of the roots 256
4.9 The general equation of the nth d e w 262
4.10 Equations with rational coefficients and symmhc group as
Galois group 267
4.1 1 Constructible regular n-gons 271
4.12 Transandena of e and r The Lindemann-Weicrstrass thcorem 277
4.13 Finite fields 287
4.14 Specs bases for finite dimensional extcmhne &Ids 2QO
4.15 Tram and norms 296
4.16 Mod p reduction 301
5 REAL POLYNOMIAL EQUATIONS AND INEQUALITIES 306
5.1 Ordered fields. Red closed fields 307
5.2 Stunn's theorem 311
5.3 Formalized Euclidean algorithon and Stm's thwm 316
5.4 Himhation proadures. Resultants 322
5.5 Deckion method for an algebraic curve 327
5.6 Tarski's thwrem 335
6 METRIC VECTOR SPACES AND THE CLASSICAL GROUPS 342
6.1 Linear functions and bilinear forms 343
6.2 Altmate forms 349
6.3 Quadratic forms and symmetric bilinear forms 354
6.4 Basic concepts of orthogonal geometry 361
6.5 Witt's canallation thcom 367
6.6 The thwm of Cartan-DieudonnC 371
6.7 Structure of the gencral linear group GLn(F) 375
6.8 Structure of orthogonal groups 382
6.9 Symplcctic geometry. The symplectic group 391
6.10 Orders of orthogonal and symplectic groups over a finite field 398
6.11 Postscript on hemitian forms and unitary geometry 401
7 ALGEBRAS OVER A FIELD 405
7.1 Dehitiou and examples of associative algebras 406
7.2 Exterior algebras. Application to determinants 411
7.3 Regular matris representations of associative algebras.
Norms and tram 422
7.4 Change of base field. Transitivity of traa and norm 426
7.5 Non-aspciative algebras. Lie and Jordan plgeb~as 430
7.6 Hunvitz' problem. Composition algebras 438
7.7 Frobenius' and Wedderburn's theorems on associative
division algebras 451
8 LATTICES AND BOOLEAN ALGEBRAS 455
8.1 Partially ordered sets and lattices 456
8.2 Distributivity and modularity 461
8.3 The theorem of Jordan-Holder-Wekind 466
8.4 The lattice of subspaces of a vector space.
Fundamental theorem of projective geometry 468
8.5 Boolean algebras 474
8.6 The Mobius function of a partially ordered set 480
Appendix 489
index 493
Since the publication of Basic Algebra I in 1974, a number of teachers and stu-
dents of the text have communicated to the author corrections and suggestions
for improvements as well as additional exercises. Many of these have been in-
corporated in this new edition. Especially noteworthy were the suggestions
b
sent by Mr. Huah Chu of National Taiwan University, Professor M a ~ Jn.
Greenberg of the University of California at Santa Cmz, Professor J. D. Reid of
Wesleyan University, Tsuneo Tamagawa of Yale University, and Professor F. D.
Veldkamp of the University of Utrecht. We are grateful to these people and
others who encouraged us to believe that we were on the right track in adopting
the point of view taken in Basic Algebra I.
Two important changes occur in the chapter on Galois theory, Chapter 4. The
first is a completely rewritten section on finite fields (section 4.13). The new ver-
sion spells out the principal results in the fonn of formal statements of theorems.
In the first edition these results were buried in the account, which was a tour de
force of brevity. In addition, we have incorporated in the text the proof of Gauss'
formula for the number N(n, q) of monic irreducible polynomials of degree n
in a finite field of q elements. In the first edition this formula appeared in an
exercise (Exercise 20, p. 145). This has now been altered to ask for N(2, q) and
xii ~refac8
N(3, q) only. The second important change in Chapter 4 is the addition of sec-
tion 4.16, "Mod p Reduction," which gives a proof due to John Tate of a theorem
of Dedekind's on the existence of certain cycles in the Galois permutation group
of the roots of an irreducible monic polynomial f(x) with integer coefficients
that can be deduced from the factorization of f(x) modulo a prime p. A number of
interesting applications of this theorem are given in the exercises at the end of
the section.
In Chapter 5 we have given a new proof of the basic elimination theorem
(Theorem 5.6). The new proof is completely elementary, and is independent of
the formal methods developed in Chapter 5 for the proof of Tariski's theorem
on elimination of quantifiers for real closed fields. Our purpose in giving the
new proof is that Theorem 5.6 serves as the main step in the proof of Hilbert's
Nullstellensatz given on pp. 424-426 of Basic Algebra 11. The change has been
made for the convenience of readers who do not wish to familiarize themselves
with the formal methods developed in Chapter 5.
At the end of the book we have added an appendix entitled "Some Topics for
Independent Study," which lists 10 such topics. There is a brief description of
each, together with some references to the literature. While some of these might
have been treated as integral parts of the text, we feel that students will benefit
more by pursuing them on their own.
The items listed account for approximately 10 pages of added text. The remain-
ing 15 or so pages added in this edition can be accounted for by local improve-
ments in the exposition and additional exercises.
The text of the second edition has been completely reset, which presented the
chore of proofreading a lengthy manuscript. This arduous task was assumed
largely by the following individuals: Huah Chu (mentioned above), Jone-Wen
Cohn of Shanghai Normal University, Florence D. Jacobson ("Florie," to whom
the book is dedicated),a nd James D. Reid (also mentioned above). We are deeply
indebted to them for their help.
Hamden, Connecticut Nathan Jacobson
November 1, 1984
Preface to the First Edition
It is more than twenty years since the author began the project of writing the
three volumes of Lectures in Abstract Algebra. The first and second of these
books appeared in 1951 and 1953 respectively, the third in 1964. In the period
which has intervened since this work was conceived-around 1950--substantial
progress in algebra has occurred even at the level of these texts. This has taken
the fonn first of all of the introduction of some basic new ideas. Notable ex-
amples are the development of category theory, which provides a useful frame-
work for a large part of mathematics, homological algebra, and applications
of model theory to algebra. Perhaps even more striking than the advent of these
ideas has been the acceptance of the axiomatic conceptual method of abstract
algebra and its pervading influence throughout mathematics. It is now taken
for granted that the methodology of algebra is an essential tool in mathematics.
On the other hand, in recent research one can observe a return to the challenge
presented by fairly concrete problems, many of which require for their solution
tools of considerable technical complexity.
Another striking change that has taken place during the past twenty yea*
especially since the Soviet Union startled the world by orbiting its "sputniks"-
has been the upgrading of training in mathematics in elementary and secondary