Introduction to Abstract Algebra

352 Pages·1975·5.81 MB·English

by  Louis W. Shapiro

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AND APPLIED MATHEMATICS
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McGRAW-HILL
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LOUIS SHAPIRO
Howard University
Introduction
to
Abstract
Algebra
This book was set in Times New Roman.
The editors were A. Anthony Arthur and David Damstra;
the production supervisor was Dennis J. Conroy.
The drawings were done by Santype Ltd.
R. R. Donnelley & Sons Company was printer and binder.
Library of Congress Cataloging in Publication Data
Shapiro, Louis, date
Introduction to abstract algebra.
(International series in pure and applied mathematics)
Bibliography: p.
I. Algebra, Abstract. I. Title.
QA162.S47 512'.02 74-11448
ISBN 0-07-056415-9
INTRODUCTION
TO
ABSTRACT
ALGEBRA
Copyright © 1975 by McGraw-Hill, Inc. All rights reserved.
Printed in the United States of America. No part of this publication may be reproduced
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher.
1 234567890 DO DO 798765
CONTENTS
Preface ix
Groups
1 1
*J•J Sets and Binary Operations
1.2 Groups, Definitions and Examples 5
1.3 Elementary Properties of Groups 7
1.4 Subgroups and Cyclic Groups 16
*J5
Functions 20
1.6 2 x 2 Matrices 26
1.7 Permutations 30
*1.8 Equivalence Relations 38
1.9 Lagrange's Theorem 41
1.10 Isomorphisms 45
*J•JJ Euclid's Algorithm and the Linear Property 51
1.12 Cyclic Groups and Direct Products 54
1.13 Homomorphisms and Normal Subgroups 59
Note: Explanations of the asterisk and dagger symbols are given in the preface.
Vi CONTENTS
V.14 The Commutator Subgroup and a Universal Mapping
Property
70
tl.15 Odds and Ends
75
V.16 Some Historical Remarks
81
2 Rings
85
2.1 Definitions and Examples 85
*t2.2 Two Important Examples
92
2.3 Matrix Rings 101
2.4 Subrings, Ideals, and Ring Homomorphisms 106
2.5 Factoring out Maximal Ideals in a Commutative Ring 116
2.6 Polynomial Rings 120
3 Vector Spaces 128
3.1 Basic Definitions 128
3.2 Bases and Dimension 134
3.3 Linear Transformations 139
3.4 Matrices 144
4 Groups Acting on Sets
155
4.1 Basic Definitions 155
4.2 Fixed Points of p-groups 159
4.3 The Burnside Counting Theorem 163
4.4 Some Applications in Group Theory 169
5 Further Results on Groups 178
5.1 Solvable Groups 179
5.2 Finitely Generated Abelian Groups 186
6 Integral Domains 192
6.1 Definitions and Quotient Fields 192
6.2 Euclidean Domains and Principal Ideal Domains 195
6.3 Unique Factorization Domains and Applications 199
6.4 Odds and Ends 207
7 Number Theory 211
7.1 Basic Results 211
7.2 Quadratic Residues 216
7.3 The Two-Square Theorem of Fermat 221
8 The Wedderburn Theorems (a Survey) 224
CONTENTS Vii
9 Group Representations (a Survey) 232
9.1 Representations 232
9.2 Characters 246
10 A Survey of Galois Theory 253
10.1 Some Field Theory 253
10.2 Geometric Constructions and Impossibility Theorems 261
10.3 Galois Theory (Preliminaries) 265
10.4 Galois Theory (Fundamental theorem) 273
Bibliography 282
Selected Answers and Occasional Hints
and Comments 287
Cross References to Elementary Mathematics 327
Notation 329
Subject Index 333

PREFACE
The emphasis in this book is on examples and exercises, and they provide much
of the motivation for the material. I have also tried to provide some historical
comment and to examine the connections between modern algebra and other
fields. (These comments and connections often appear in the exercises.) In
particular I have tried to point out the connections with elementary mathematics
as they occur. A special cross-reference index has also been included for this
purpose.
The basic plan of the book is as follows: The first two chapters cover
groups, homomorphisms, and rings. Set theory, functions, equivalence rela-
tions, the integers, and the complex numbers are developed when needed, rather
than in a chapter. A trimester or semester course can cover the first two
chapters, especially if the sections marked t are omitted. The rest of the book
is devoted to various topics which might be covered in a second trimester or
semester. The main options are:
1 Groups acting on sets
2 Integral domains, number theory
3 Vector spaces (Secs. 3.1 and 3.2), further group theory (Sec. 5.1), field theory
4 Vector spaces, Wedderburn theorems, group representations
5 Wedderburn theorems