Table Of Content

A BOOK OF ABSTRACT ALGEBRA Second Edition Charles C. Pinter Professor of Mathematics Bucknell University Dover Publications, Inc., Mineola, New York Copyright Copyright © 1982, 1990 by Charles C. Pinter All rights reserved. Bibliographical Note This Dover edition, first published in 2010, is an unabridged republication of the 1990 second edition of the work originally published in 1982 by the McGraw-Hill Publishing Company, Inc., New York. Library of Congress Cataloging-in-Publication Data Pinter, Charles C, 1932– A book of abstract algebra / Charles C. Pinter. — Dover ed. p. cm. Originally published: 2nd ed. New York : McGraw-Hill, 1990. Includes bibliographical references and index. ISBN-13: 978-0-486-47417-5 ISBN-10: 0-486-47417-8 1. Algebra, Abstract. I. Title. QA162.P56 2010 512′.02—dc22 2009026228 Manufactured in the United States by Courier Corporation 47417803 www.doverpublications.com To my wife, Donna, and my sons, Nicholas, Marco, Andrés, and Adrian CONTENTS* Preface Chapter 1 Why Abstract Algebra? History of Algebra. New Algebras. Algebraic Structures. Axioms and Axiomatic Algebra. Abstraction in Algebra. Chapter 2 Operations Operations on a Set. Properties of Operations. Chapter 3 The Definition of Groups 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 Uniqueness of Identity and Inverses. Properties of Inverses. Direct Product of Groups. Chapter 5 Subgroups Definition of Subgroup. Generators and Defining Relations. Cay ley Diagrams. Center of a Group. Group Codes; Hamming Code. Chapter 6 Functions Injective, Surjective, Bijective Function. Composite and Inverse of Functions. Finite-State Machines. Automata and Their Semigroups. Chapter 7 Groups of Permutations Symmetric Groups. Dihedral Groups. An Application of Groups to Anthropology. Chapter 8 Permutations of a Finite Set Decomposition of Permutations into Cycles. Transpositions. Even and Odd Permutations. Alternating Groups. Chapter 9 Isomorphism The Concept of Isomorphism in Mathematics. Isomorphic and Nonisomorphic Groups. Cayley’s Theorem. Group Automorphisms. Chapter 10 Order of Group Elements Powers/Multiples of Group Elements. Laws of Exponents. Properties of the Order of Group Elements. Chapter 11 Cyclic Groups Finite and Infinite Cyclic Groups. Isomorphism of Cyclic Groups. Subgroups of Cyclic Groups. Chapter 12 Partitions and Equivalence Relations Chapter 13 Counting Cosets Lagrange’s Theorem and Elementary Consequences. Survey of Groups of Order ≤ 10. Number of Conjugate Elements. Group Acting on a Set. Chapter 14 Homomorphisms Elementary Properties of Homomorphisms. Normal Subgroups. Kernel and Range. Inner Direct Products. Conjugate Subgroups. Chapter 15 Quotient Groups Quotient Group Construction. Examples and Applications. The Class Equation. Induction on the Order of a Group. Chapter 16 The Fundamental Homomorphism Theorem 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 Commutative Rings. Unity. Invertibles and Zero-Divisors. Integral Domain. Field. Chapter 18 Ideals and Homomorphisms Chapter 19 Quotient Rings 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 Ordered Integral Domains. Well-ordering. Characterization of Up to Isomorphism. Mathematical Induction. Division Algorithm. Chapter 22 Factoring into Primes Ideals of . Properties of the GCD. Relatively Prime Integers. Primes. Euclid’s Lemma. Unique Factorization. Chapter 23 Elements of Number Theory (Optional) Properties of Congruence. Theorems of Fermât 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 Motivation and Definitions. Domain of Polynomials over a Field. Division Algorithm. Polynomials in Several Variables. Fields of Polynomial Quotients. Chapter 25 Factoring Polynomials Ideals of F[x]. Properties of the GCD. Irreducible Polynomials. Unique factorization. Euclidean Algorithm. Chapter 26 Substitution in Polynomials Roots and Factors. Polynomial Functions. Polynomials over . Eisenstein’s Irreducibility Criterion. Polynomials over the Reals. Polynomial Interpolation. Chapter 27 Extensions of Fields Algebraic and Transcendental Elements. The Minimum Polynomial. Basic Theorem on Field Extensions. Chapter 28 Vector Spaces Elementary Properties of Vector Spaces. Linear Independence. Basis. Dimension. Linear Transformations. Chapter 29 Degrees of Field Extensions Simple and Iterated Extensions. Degree of an Iterated Extension. Fields of Algebraic Elements. Algebraic Numbers. Algebraic Closure. Chapter 30 Ruler and Compass Constructible Points and Numbers. Impossible Constructions. Constructible Angles and Polygons. Chapter 31 Galois Theory: Preamble Multiple Roots. Root Field. Extension of a Field. Isomorphism. Roots of Unity. Separable Polynomials. Normal Extensions. Chapter 32 Galois Theory: The Heart of the Matter Field Automorphisms. The Galois Group. The Galois Correspondence. Fundamental Theorem of Galois Theory. Computing Galois Groups. Chapter 33 Solving Equations by Radicals Radical Extensions. Abelian Extensions. Solvable Groups. Insolvability of the Quin tic. Appendix A Review of Set Theory Appendix B Review of the Integers Appendix C Review of Mathematical Induction Answers to Selected Exercises Index * Italic headings indicate topics discussed in the exercise sections. 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 conventional 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 algebra, 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 meaningfulness of algebraic concepts, by tracing these concepts to their origins in classical algebra and at the same time exploring their connections with other parts of mathematics, especially 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 intuitive content of the 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 definition, theorem, proof, corollary, example. In my experience, that kind of format encourages some students to believe that mathematical concepts have a merely conventional character, and may encourage rote memorization. Instead, each chapter has the form of a discussion with the student, with the accent on explaining and motivating. In an effort to avoid fragmentation of the subject matter into loosely related definitions and results, each chapter is built around a central theme and remains anchored to this focal point. In the later chapters especially, this focal point is a specific application or use. Details of every topic are then woven into the general discussion, so as to keep a natural flow of ideas running through each chapter. The arrangement of topics is designed to avoid tedious proofs and long-winded explanations. Routine arguments are worked into the discussion whenever this seems natural and appropriate, and proofs to theorems are seldom more than a few lines long. (There are, of course, a few exceptions to this.) Elementary background material is filled in as it is needed. For example, a brief chapter on functions precedes the discussion of permutation groups, and a chapter on equivalence relations and partitions paves the way for Lagrange’s theorem. This book addresses itself especially to the average student, to enable him or her to learn and understand as much algebra as possible. In scope and subject-matter coverage, it is no different from many other standard texts. It begins with the promise of demonstrating the unsolvability of the quintic and ends with that promise fulfilled. Standard topics are discussed in their usual order, and many advanced and peripheral subjects are introduced in the exercises, accompanied by ample instruction and commentary. I have included a copious supply of exercises—probably more exercises than in other books at this level. They are designed to offer a wide range of experiences to students at different levels of ability. There is some novelty in the way the exercises are organized: at the end of each chapter, the exercises are grouped into exercise sets, each set containing about six to eight exercises and headed by a descriptive title. Each set touches upon an idea or skill covered in the chapter. The first few exercise sets in each chapter contain problems which are essentially computational or manipulative. Then, there are two or three sets of simple proof-type questions, which require mainly the ability to put together definitions and results with understanding of their meaning. After that, I have endeavored to make the exercises more interesting by arranging them so that in each set a new result is proved, or new light is shed on the subject of the chapter. As a rule, all the exercises have the same weight: very simple exercises are grouped together as parts of a single problem, and conversely, problems which require a complex argument are broken into several subproblems which the student may tackle in turn. I have selected mainly problems which have intrinsic relevance, and are not merely drill, on the premise that this is much more satisfying to the student. CHANGES IN THE SECOND EDITION During the seven years that have elapsed since publication of the first edition of A Book of Abstract Algebra, I have received letters from many readers with comments and suggestions. Moreover, a number of reviewers have gone over the text with the aim of finding ways to increase its effectiveness and appeal as a teaching tool. In preparing the second edition, I have taken account of the many suggestions that were made, and of my own experience with the book in my classes. In addition to numerous small changes that should make the book easier to read, the following major changes should be noted: EXERCISES Many of the exercises have been refined or reworded—and a few of the exercise sets reorganized—in order to enhance their clarity or, in some cases, to make them more mathematically interesting. In addition, several new exericse sets have been included which touch upon applications of algebra and are discussed next: APPLICATIONS The question of including “applications” of abstract algebra in an undergraduate course (especially a one-semester course) is a touchy one. Either one runs the risk of making a visibly weak case for the applicability of the notions of abstract algebra, or on the other hand—by including substantive applications—one may end up having to omit a lot of important algebra. I have adopted what I believe is a reasonable compromise by adding an elementary discussion of a few application areas (chiefly aspects of coding and automata theory) only in the exercise sections, in connection with specific exercise. These exercises may be either stressed, de-emphasized, or omitted altogether. PRELIMINARIES It may well be argued that, in order to guarantee the smoothe flow and continuity of a course in abstract algebra, the course should begin with a review of such preliminaries as set theory, induction and the properties of integers. In order to provide material for teachers who prefer to start the course in this fashion, I have added an Appendix with three brief chapters on Sets, Integers and Induction, respectively, each with its own set of exercises. SOLUTIONS TO SELECTED EXERCISES A few exercises in each chapter are marked with the symbol #. This indicates that a partial solution, or sometimes merely a decisive hint, are given at the end of the book in the section titled Solutions to Selected Exercises.