Table Of Content

oako stract r SECC)1'0E DITION • \ \ 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