Abstract Algebra: An Introduction To Groups, Rings And Fields

by Clive Reis

Overview

This book is appropriate for second to fourth year undergraduates. In addition to the material traditionally taught at this level, the book contains several applications: Polya-Burnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes and a classification of the finite groups of isometries of the plane and the finite rotation groups in Euclidean 3-space. It is hoped that these applications will help the reader achieve a better grasp of the rather abstract ideas presented and convince him/her that pure mathematics, in addition to having an austere beauty of its own, can be applied to solving practical problems.Considerable emphasis is placed on the algebraic system consisting of congruence classes mod n under the usual operations of addition and multiplication. The reader is thus introduced — via congruence classes — to the idea of cosets and factor groups. This enables the transition to cosets and factor objects in a more abstract setting to be relatively painless. The chapters dealing with applications help to reinforce the concepts and methods developed in the context of more down-to-earth problems.Most introductory texts in abstract algebra either avoid cosets, factor objects and homomorphisms completely or introduce them towards the end of the book. In this book, these topics are dealt with early on so that the reader has at his/her disposal the tools required to give elegant proofs of the fundamental theorems. Moreover, homomorphisms play such a prominent role in algebra that they are used in this text wherever possible, even if there are alternative methods of proof.

Product Details

ISBN-13:	9789814340281
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/10/2011
Pages:	508
Product dimensions:	6.00(w) x 8.90(h) x 1.10(d)

Table of Contents

Preface vii

List of Tables ix

List of Figures xi

Chapter 1 Logic and Proofs 1

1.1 Introduction 1

1.2 Statements, Connectives and Truth Tables 2

1.3 Relations Between Statements 6

1.4 Quantifiers 7

1.5 Methods of proof 10

1.6 Exercises 13

Chapter 2 Set Theory 17

2.1 Definitions 17

2.2 Relations Between Sets 19

2.3 Operations Defined on Sets - Or New sets from Old 20

2.4 Exercises 24

Chapter 3 Cartesian Products, Relations, Maps and Binary Operations 29

3.1 Introduction 29

3.2 Cartesian Products 29

3.3 Maps 37

3.4 Binary Operations 46

3.5 Exercises 53

Chapter 4 The Integers 59

4.1 Introduction 59

4.2 Elementary Properties 59

4.3 Divisibility 67

4.4 The Fundamental Theorem of Arithmetic 73

4.5 The Algebraic System (Z_n,+, ) and Congruences 76

4.6 Congruences in Z and Equations in Z_n 85

4.7 Exercises 91

Chapter 5 Groups 97

5.1 Introduction 97

5.2 Definitions and Elementary Properties 98

5.3 Alternative Axioms for Groups 106

5.4 Subgroups 108

5.5 Cyclic Groups 115

5.6 Exercises 120

Chapter 6 Further Properties of Groups 127

6.1 Introduction 127

6.2 Cosets 127

6.3 Isomorphisms and Homomorphisms 135

6.4 Normal Subgroups and Factor Groups 142

6.5 Direct Products of Groups 154

6.6 Exercises 158

Chapter 7 The Symmetric Groups 165

7.1 Introduction 165

7.2 The Cayley Representation Theorem 165

7.3 Permutations as Products of Disjoint Cycles 167

7.4 Odd and Even Permutations 172

7.5 Conjugacy Classes of a Group 178

7.6 Exercises 183

Chapter 8 Rings, Integral Domains and Fields 187

8.1 Rings 187

8.2 Homomorphisms, Isomorphisms and Ideals 194

8.3 Isomorphism Theorems 199

8.4 Direct Sums of Rings 201

8.5 Integral Domains and Fields 206

8.6 Embedding an Integral Domain in a Field 212

8.7 The Characteristic of an Integral Domain 215

8.8 Exercjses 218

Chapter 9 Polynomial Rings 229

9.1 Introduction 229

9.2 Definitions and Elementary Properties 230

9.3 The Division Algorithm and Applications 234

9.4 Irreducibility and Factorization of Polynomials 241

9.5 Polynomials Over More Familiar Fields 247

9.6 Factor Rings of the Form F[x]/(g(x)), F a Field 255

9.7 Exercises 263

Chapter 10 Field Extensions 269

10.1 Introduction 269

10.2 Definitions and Elementary Results 269

10.3 Algebraic and Transcendental Elements 275

10.4 Algebraic Extensions 278

10.5 Finite Fields 286

10.6 Exercises 291

Chapter 11 Latin Squares and Magic Squares 297

11.1 Latin Squares 297

11.2 Magic Squares 303

11.3 Exercises 306

Chapter 12 Group Actions, the Class Equation and the Sylow Theorems 309

12.1 Group Actions 309

12.2 The Class Equation of a Finite Group 314

12.3 The Sylow Theorems 315

12.4 Applications of the Sylow Theorems 321

12.5 Exercises 335

Chapter 13 Isometries 341

13.1 Isometries of Rⁿ 341

13.2 Finite Subgroups of E(2) 345

13.3 The Platonic Solids 348

13.4 Rotations in R³ 353

13.5 Exercises 359

Chapter 14 Polya-Burnside Enumeration 363

14.1 Introduction 363

14.2 A Theorem of Polya 366

14.3 Exercises 373

Chapter 15 Group Codes 377

15.1 Introduction 377

15.2 Definitions and Notation 379

15.3 Group Codes 384

15.4 Construction of Group Codes 388

15.5 At the Receiving End 390

15.6 Nearest Neighbor Decoding for Group Codes 392

15.7 Hamming Codes 397

15.8 Exercises 399

Chapter 16 Polynomial Codes 405

16.1 Definitions and Elementary Results 405

16.2 BCH Codes 412

16.3 Exercises 420

Appendix A Rational, Real and Complex Numbers 423

A.1 Introduction 423

A.2 The Real and Rational Number Systems 424

A.3 Decimal Representation of Rational Numbers 427

A.4 Complex Numbers 428

A.5 Polar Form of a Complex Number 432

A.6 Exercises 440

Appendix B Linear Algebra 445

B.1 Vector Spaces 445

B.2 Linear Transformations 452

B.3 Inner Product Spaces 462

B.4 Orthogonal Linear Transformations and Orthogonal Matrices 468

B.5 Determinants 471

B.6 Eigenvalues and Eigenvectors 480

B.7 Exercises 482

Index 487