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  • INTRODUCTIONTO ABSTRACT

    ALGEBRA

    Elbert A. WalkerNew Mexico State UniversityLas Cruces, New Mexico, USA

  • c1998 by Elbert A. Walker, New Mexico State University.All rights reserved. No part of this document may be reproduced, stored in a re-trieval system, or transcribed, in any form or by any means electronic, mechanical,photocopying, recording, or otherwise without the prior written permission of theauthor.

    Revision of: Introduction to Abstract Algebra by Elbert A. Walker; c1987 by RandomHouse, Inc.

    This document was produced by Scientific WorkPlace TM.

  • Contents

    1 Sets 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Sets and Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . 151.6 The Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    2 Groups 252.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 252.2 Subgroups and Cosets . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . 522.5 The Groups Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.6 Direct Products and Semi-direct Products . . . . . . . . . . . . . 622.7 Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . 69

    3 Vector Spaces 793.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 793.2 Homomorphisms of Vector Spaces . . . . . . . . . . . . . . . . . . 883.3 Linear Independence and Bases . . . . . . . . . . . . . . . . . . . 943.4 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

    4 Rings and Modules 1114.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 1114.2 Homomorphisms and Quotient Rings . . . . . . . . . . . . . . . . 1184.3 Field of Quotients of an Integral Domain . . . . . . . . . . . . . . 1264.4 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . 1304.5 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.6 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.7 Modules over Principal Ideal Domains . . . . . . . . . . . . . . . 157

    iii

  • iv CONTENTS

    5 Linear Transformations 1695.1 Linear Transformations and Matrices . . . . . . . . . . . . . . . . 1695.2 The Rational Canonical Form for Matrices . . . . . . . . . . . . . 1775.3 Eigenvectors and Eigenvalues; the Jordan Canonical Form[Eigenvectors

    and Eigenvalues] . . . . . . . . . . . . . . . . . . . . . . . . . . . 1875.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1975.5 Equivalence of Matrices . . . . . . . . . . . . . . . . . . . . . . . 2125.6 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

    6 Fields 2376.1 Subfields and Extension Fields . . . . . . . . . . . . . . . . . . . 2376.2 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2436.3 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2516.4 Solvability by Radicals . . . . . . . . . . . . . . . . . . . . . . . . 261

    7 Topics from Group Theory 2697.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2697.2 The Jordan-Hlder Theorem . . . . . . . . . . . . . . . . . . . . 2697.3 The Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 2727.4 Solvable and Nilpotent Groups . . . . . . . . . . . . . . . . . . . 2757.5 The Wedderburn Theorem for Finite Division Rings . . . . . . . 2807.6 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . 281

    8 Topics in Ring Theory 2838.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2838.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . . . 2848.3 The Hilbert Basis Theorem . . . . . . . . . . . . . . . . . . . . . 2898.4 The Lasker-Noether Decomposition Theorem . . . . . . . . . . . 2918.5 Semisimple Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

    Appendix: Zorns Lemma 305

    Bibliography 311

  • Preface

    In teaching a beginning course in abstract algebra, one must suppress the urgeto cover a lot of material and to be as general as possible. The diffi culties inteaching such a course are pedagogical, not mathematical. The subject matteris abstract, yet it must be kept meaningful for students meeting abstractness forperhaps the first time. It is better for a student to know what a theorem saysthan to be able to quote it, produce a proof of it detail by detail, and not havethe faintest notion of what its all about. However, careful attention must bepaid to rigor, and sloppy thinking and incoherent writing cannot be tolerated.But rigor should be flavored with understanding. Understanding the content ofa theorem is as important as being able to prove it. I have tried to keep thesethings in mind while writing this book.The specific subject matter chosen here is standard, and the arrangement of

    topics is not particularly bizarre. In an algebra course, I believe one should geton with algebra as soon as possible. This is why I have kept Chapter 1 to a bareminimum. I didnt want to include it at all, but the material there is absolutelyessential for Chapter 2, and the studentsknowledge of it is apt to be a bit hazy.Other bits of set theorywill be expounded upon as the need arises. ZornsLemma and some of its applications are discussed in the Appendix.Groups are chosen as the first algebraic system to study for several reasons.

    The notion is needed in subsequent topics. The axioms for a group are simplerthan those for the other systems to be studied. Such basic notions as homomor-phisms and quotient systems appear in their simplest and purest forms. Thestudent can readily recognize many old friends as abstract groups in concretedisguise.The first topic slated for a thorough study is that of vector spaces. It is

    this material that is most universally useful, and it is important to presentit as soon as is practical. In fact, the arrangement of Chapters 3 through 5is a compromise between mathematical effi ciency and getting the essentials oflinear algebra done. Chapter 4 is where it is because its results are beautifullyapplicable in Chapter 5 and contain theorems one would do anyway. Besides,there are some nice applications of Chapter 3 in Chapter 4. Chapter 6 is basicand should not be slighted in favor of 7 or 8. A feature of Chapter 7 is analgebraic proof of the fundamental theorem of algebra.There are many exercises in this book. Mathematics is not a spectator sport,

    and is best learned by doing. The exercises are provided so that students cantest their knowledge of the material in the body of the text, practice concoctingproofs on their own, and pick up a few additional facts.

  • vi CONTENTS

    There is no need here to extol the virtues of the abstract approach, andthe importance of algebra to the various areas of mathematics. They are wellknown to the professional, and will become fully appreciated by the studentonly through experience.

    Elbert A. WalkerLas Cruces, New MexicoMarch, 1986

  • Chapter 1

    Sets

    1.1 Introduction

    The purpose of this book is to present some of the basic properties of groups,rings, fields, and vector spaces. However, some preliminaries are necessary.There are some facts about sets, mappings, equivalence relations, and the likethat will be used throughout the text, and which are indispensable. This chapterpresents those facts. It is not meant to be an introduction to set theory. Theamount of material is kept to a minimum just what is needed to get startedon algebra.A few fundamental facts concerning the integers will be assumed. These are

    spelled out in Section 1.6.

    1.2 Sets and Subsets

    Our approach to sets is the naive or intuitive one. An axiomatic approach tosets here would lead us too far afield and would contribute nothing toward theunderstanding of the basic properties of a field, for example.By the word set, we mean any collection of objects. The objects in the

    collection are called the elements of the set. Sets will usually be denoted bycapital letters. If s is an element of a set S, we write s S, and if s is not anelement of S, we write s / S. For any object s, either s S or s / S. For theset Z of all integers, we have 5 Z, but 1/2 / Z. A set is determined by itselements. That is, two sets are the same if they have the same elements. Thustwo sets S and T are equal, written S = T , if s S implies s T , and t Timplies t S.To specify a set, we must tell what its elements are, and this may be done

    in various ways. One way is just to write out an ordinary sentence which doesthat. For example, we could say S is the set whose elements are the integers1, 2, and 3. A shorthand has been developed for this sort of thing, however.S = {1, 2, 3} means that S is the set whose elements are 1, 2, and 3. Thus,

    1

  • 2 CHAPTER 1. SETS

    one way to define a set is to list its elements and put braces around the list.If S is a big set, this procedure can be cumbersome. For example, it would betiresome to describe the set of positive integers less that 1,000,000 in this way.The sentence we have just written would be more effi cient. However, there isa convention that is universally used that is most convenient. Suppose S is aset, and that it is the set of all elements s which satisfy some property P . ThenS = {s : s satisfies property P} means that S is the set of all elements s whichenjoy property P . For example,