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A BOOKO F
ABSTRAACLTG EBRA
Second Edition
Charles C. Pinter
Professor of Mathematics
Bucknell University
McGraw-Hill Publishing Company
New York St. Louis San Francisco Auckland Bogota Caracas Hamburg
Lisbon London Madrid Mexico Milan Montreal New Delhi Oklahoma City
Paris San Juan Sao Paulo Singapore Sydney Tokyo Toronto
This book was set in Times Roman.
The editors were Robert A. Weinstein and Margery Luhrs;
the production supervisor was Salvador Gonzales.
The cover was designed by Hermann Strohbach.
R. R. Donnelley & Sons Company was printer and binder.
A BOOK OF ABSTRACT ALGEBRA
Copyright © 1990, 1982 by McGraw-Hill, Inc. All rights reserved. Printed in the
United States of America. Except as permitted under the United States Copyright
Act of 1976, no part of this publication may be reproduced or distributed in any
form or by any means, or stored in a data base or retrieval system, without the
prior written permission of the publisher.
1 2 3 4 5 6 7 8 9 0 DOC DOC 8 9 4 3 2 1 0 9
□ □ □
ISBN- 075-�38-b
Library of Congress Cataloging-in-Publication Data
Pinter, Charles C., (date).
A book of abstract algebra / Charles C. Pinter. -2nd ed.
p. cm. - (International series in pure and applied
mathematics) (McGraw-Hill series in higher mathematics)
Includes index.
ISBN 0-07-050138-6
1. Algebra, Abstract. I. Title. II. Series. Ill. Series:
McGraw-Hill series in higher mathematics.
QA162.P56 1990
512'.02-dc20 39:.35355
To my wife, Donna,
and my sons,
Nicholas, Marco,
Andres, and Adrian
CONTENTS*
Preface
xv
Chapter 1 Why Abstract Algebra?
1
History of Algebra. New Algebras. Algebraic
Structures. Axioms and Axiomatic Algebra.
Abstraction in Algebra.
Chapter 2 Operations
19
Operations on a Set. Properties of Operations.
Chapter 3 The Definition of Groups 2
5
Groups. Examples of Infinite and Finite Groups.
Examples of Abelian and Nonabelian Groups.
Group Tables.
Theory of Coding: Maximum-Likelihood Decoding.
Chapter 4 Elementary Properties of Groups 36
Uniqueness of Identity and Inverses. Properties
of Inverses.
Direct Product of Groups.
Chapter 5 Subgroups
44
Definition of Subgroup. Generators and Defining
Relations.
Cayley Diagrams. Center of a Group. Group Codes;
Hamming Code.
* Italic headings indicate topics discussed in the exercise sections.
ix
X CONTENTS
Chapter 6 Functions 56
Injective, Surjective, Bijective Function. Composite
and Inverse of Functions.
Finite-State Machines. Automata and Their Semigroups.
Chapter 7 Groups of Permutations
69
Symmetric Groups. Dihedral Groups.
An Application of Groups to Anthropology.
Chapter 8 Permutations of a Finite Set
80
Decomposition of Permutations into Cycles.
Transpositions. Even and Odd Permutations.
Alternating Groups.
Chapter 9 Isomorphism
90
The Concept of Isomorphism in Mathematics.
Isomorphic and Nonisomorphic Groups.
Cayley's Theorem.
Group Automorphisms.
Chapter 10 Order of Group Elements 103
Powers/Multiples of Group Elements. Laws of
Exponents. Properties of the Order of Group
Elements.
Chapter 11 Cyclic Groups 112
Finite and Infinite Cyclic Groups. Isomorphism
of Cyclic Groups. Subgroups of Cyclic Groups.
Chapter 12 Partitions and Equivalence Relations 119
Chapter 13 Counting Cosets 126
'
Lagranges Th�orem and Elementary Consequences.
Survey of Groups of Order ,:;;;; 10.
Number of Conjugate Elements. Group Acting on a Set.
Chapter 14 Homomorphisms
136
Elementary Properties of Homomorphisms. Normal
Subgroups. Kernel and Range.
Inner Direct Products. Conjugate Subgroups.
CONTENTS xi
Chapter 15 Quotient Groups
147
Quotient Group Construction. Examples and
· Applications.
The Class Equation. Induction on the Order of a Group.
Chapter 16 The Fundamental Homomorphism
Theorem
r 157
Fundamental Homomorphism Theorem and Some
Consequences.
The Isomorphism Theorems. The Correspondence Theorem.
Cauchy's Theorem. Sylow Subgroups. Sylow's Theorem.
Decomposition Theorem for Finite Abelian Groups.
Chapter 17 Rings: Definitions and
Elementary Properties
169
Commutative Rings. Unity. Invertibles and
Zero-Divisors. Integral Domain. Field.
Chapter 18 Ideals and Homomorphisms
181
Chapter 19 Quotient Rings
190
Construction of Quotient Rings. Examples.
Fundamental Homomorphism Theorem and Some
Consequences. Properties of Prime and Maximal
Ideals.
Chapter 20 Integral Domains
Characteristic of an Integral Domain. Properties
of the Characteristic. Finite Fields. Construction
of the Field of Quotients.
Chapter 21 The Integers
208
Ordered Integral Domains. Well-ordering.
Characterization of 7L Up to Isomorphism. Mathematical
Induction. Division Algorithm.
Chapter 22 Factoring into Primes
217
Ideals of 7L. Properties of the GCD. Relatively
Prime Integers. Primes. Euclid's Lemma. Unique
Factorization.
xii CONTENTS
Chapter 23 Elements of Number Theory (Optional)
226
Properties of Congruence. Theorems of Fermat and
Euler. Solutions of Linear Congruences. Chinese
Remainder Theorem.
Wilson's Theorem and Consequences. Quadratic Residues.
The Legendre Symbol. Primitive Roots.
Chapter 24 Rings of Polynomials
240
Motivation and Definitions. Domain of Polynomials
over a Field. Division Algorithm.
Polynomials in Several Variables. Fields of Polynomial Quotients.
Chapter 25 Factoring Polynomials
251
F[x].
Ideals of Properties of the GCD. Irreducible
Polynomials. Unique factorization.
Euclidean Algorithm.
Chapter 26 Substitution in Polynomials
258
Roots and Factors. Polynomial Functions.
Polynomials over Q. Eisenstein's Irreducibility
Criterion. Polynomials over the Reals. Polynomial
Interpolation.
Chapter 27 Extensions of Fields
270
Algebraic and Transcendental Elements. The
Minimum Polynomial. Basic Theorem on Field
Extensions.
Chapter 28 Vector Spaces
282
Elementary Properties of Vector Spaces. Linear
Independence. Basis. Dimension. Linear
Transformations.
Chapter 29 Degrees of Field Extensions
292
Simple and Iterated Extensions. Degree of an Iterated
Extension.
Fields of Algebraic Elements. Algebraic Numbers. Algebraic
Closure.
CONTENTxSii i
Chap3t0e Rru ler and Compass
301
Constructible Points and Numbers. Impossible
Constructions.
Constructible Angles and Polygons.
Chap3t1e Gra lois Theory: Preamble
311
Multiple Roots. Root Field. Extension of a Field.
Isomorphism.
Roots of Unity. Separable Polynomials. Normal Extensions.
Chap3t2eGr a lois Theory: The Heart of the Matter
323
Field Automorphisms. The Galois Group. The
Galois Correspondence. Fundamental Theorem of
Galois Theory.
Computing Galois Groups.
Chap3t3e Sro lving Equations by Radicals
334
Radical Extensions. Abelian Extensions. Solvable
Groups. Insolvability of the Quintic.
AppenAd Riexvi ew of Set Theory
345
,
AppenBd Riexvie w of the Integers
349
AppenCd iRexvi,ew of Mathematical Induction
Answers to Selected Exercises
353
Index
381
PREFACE
Once, when I was a student struggling to understand modern algebra, I
was told to view this subject as an intellectual chess game, with conven
tional moves and prescribed rules of play. I was ill served by this bit of
extemporaneous advice, and vowed never to perpetuate the falsehood
that mathematics is purely--or primarily-a formalism. My pledge has
strongly influenced the shape and style of this · book.
While giving due emphasis to the deductive aspect of modern alge
bra, I have endeavored here to present modern algebra as a lively branch
of mathematics, having considerable imaginative appeal and resting on
some firm, clear, and familiar intuitions. I have devoted a great deal of
attention to bringing out the of algebraic concepts, by
meaningfulness
tracing these concepts to their origins in classical algebra and at the same
time exploring their connections with other parts of mathematics, espe
cially geometry, number theory, and aspects of computation and equation
solving.
In an introductory chapter entitled Why Abstract Algebra?, as well as
in numerous historical asides, concepts of abstract algebra are traced to
the historic .context in which they arose. I have attempted to show that
they arose without artifice, as a natural response to particular needs, in
the course of a natural process of evolution. Furthermore, I have
endeavored to bring to light, explicitly, the of the
intuitive content
algebraic concepts used in this book. Concepts are more meaningful to
students when the students are able to represent those concepts in their
minds by clear and familiar mental images. Accordingly, the process of
concrete concept-formation is developed with care throughout this book.
I have deliberately avoided a rigid conventional format, with its
succession of In my ex
definition, theorem, proof, corollary, example.
perience, that kind of format encourages some students to believe that
mathematical concepts have a merely conventional character, and may
xv
