Abstract Algebra: Introduction To Groups, Rings And Fields With Applications (Second Edition)

by Clive Reis, Stuart A Rankin

Overview

This second edition covers essentially the same topics as the first. However, the presentation of the material has been extensively revised and improved. In addition, there are two new chapters, one dealing with the fundamental theorem of finitely generated abelian groups and the other a brief introduction to semigroup theory and automata.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, semigroups and automata. 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 the 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 to be relatively painless.In this book, cosets, factor objects and homomorphisms are introduced 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.

Product Details

ISBN-13:	9789814730549
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/18/2016
Pages:	576
Sales rank:	702,854
Product dimensions:	6.00(w) x 8.90(h) x 1.20(d)

Table of Contents

Preface to the Second Edition v

Preface to the First Edition vii

List of Figures ix

List of Tables xi

1 Logic and Proofs 1

1.1 Introduction 1

1.2 Statement, Connectives and Truth Tables 2

1.3 Relations between Statements 5

1.4 Quantifiers 6

1.5 Methods of Proof 9

1.6 Exercises 11

2 Set Theory 13

2.1 Initial Concepts 13

2.2 Relations between Sets 15

2.3 Operations Defined on Sets -or New Sets from Old 16

2.4 Exercises 19

3 Cartesian Products, Relations, Maps and Binary Operations 23

3.1 Introduction 23

3.2 Cartesian Product 23

3.3 Maps 29

3.4 Binary Operations 37

3.5 Exercises 51

4 The Integers 55

4.1 Introduction 55

4.2 Elementary Properties 55

4.3 Divisibility 61

4.4 The Fundamental Theorem of Arithmetic 67

4.5 Congruence Modulo n and the Algebraic System (Z_n,+, ) 69

4.6 Linear Congruences in Z and Linear Equations in Z_n 75

4.7 Exercises 80

5 Groups 85

5.1 Introduction 85

5.2 Definitions and Elementary Properties 86

5.3 Alternative Axioms for Groups 93

5.4 Subgroups 96

5.5 Cyclic Groups 109

5.6 Exercises 113

6 Further Properties of Groups 117

6.1 Introduction 117

6.2 Cosets 117

6.3 Isomorphisms and Homomorphisms 123

6.4 Normal Subgroups and Factor Groups 134

6.5 Direct Product of Groups 145

6.6 Exercises 152

7 The Symmetric Groups 157

7.1 Introduction 157

7.2 The Cayley Representation Theorem 157

7.3 Permutations as Products of Disjoint Cycles 159

7.4 Even and Odd Permutations o 166

7.5 The Simplicity of A_n176

7.6 Exercises 180

8 Rings, Integral Domains and Fields 183

8.1 Rings 183

8.2 Ring Homomorphisms, Ring Isomorphisms, and Ideals 192

8.3 Isomorphism Theorems 198

8.4 Direct Product and Direct Sum of Rings 199

8.5 Principal Ideal Domains and Unique Factorization Domains 207

8.6 Embedding an Integral Domain in a Field 211

8.7 The Characteristic, of an Integral Domain 214

8.8 Exercises 217

9 Polynomial Rings 225

9.1 Introduction 225

9.2 The Ring of Power Series with Coefficients in a Commutative Unital Ring R 226

9.3 The Ring of Polynomials with Coefficients in a Commutative Unital Ring 228

9.4 The Division Algorithm and Applications 234

9.5 Irreducibility and Factorization of Polynomials 240

9.6 Polynomials over Q 245

9.7 Irreducible Polynomials over R and C 250

9.8 Quotient Rings of the Form F[x]/(f), F a Field 253

9.9 Exercises 259

10 Field Extensions 263

10.1 Introduction 263

10.2 Definitions and Elementary Results 263

10.3 Algebraic and Transcendental Elements 267

10.4 Algebraic Extensions 270

10.5 Finite Fields 276

10.6 Exercises 284

11 Latin Squares and Magic Squares 289

11.1 Introduction 289

11.2 Magic Squares 294

11.3 Exercises 296

12 Group Actions, the Class Equation, and the Sylow Theorems 299

12.1 Group Actions 299

12.2 The Class Equation of a Finite Group 303

12.3 The Sylow Theorems 304

12.4 Applications of the Sylow Theorems 307

12.5 Exercises 322

13 Finitely Generated Abelian Groups 327

13.1 Introduction and Preliminary Results 327

13.2 Direct Sum of Abelian Groups 330

13.3 Free Abelian Groups 332

13.4 Finite Abelian Groups 336

13.5 The Structure of the Group of Units of Z_n 340

13.6 Exercises 345

14 Semigroups and Automata 347

14.1 Introduction 347

14.2 Semigroups 347

14.3 The Semigroup of Relations on a Set 364

14.4 Green's Relations 367

14.5 Semigroup Actions 377

14.6 Automata Theory 385

14.7 Exercises 399

15 Isometries 407

15.1 Isometrics of Rⁿ 407

15.2 Finite Subgroups of the Isometry Group of Rⁿ 412

15.3 The Classification of the Finite Subgroups of SO(3) 416

15.4 The Platonic Solids 421

15.5 Exercises 428

16 Pólya-Burnside Enumeration 431

16.1 Introduction 431

16.2 A Theorem of Pólya 434

16.3 Enumeration Examples 436

16.4 Exercises 440

17 Group Codes 445

17.1 Introduction 445

17.2 Group Codes 451

17.3 Construction of Group Codes 454

17.4 At the Receiving End 457

17.5 Nearest Neighbor Decoding for Group Codes 457

17.6 Hamming Codes 460

17.7 Exercise 463

18 Polynomial Codes 467

18.1 Definitions and Elementary Results 467

18.2 BCH Codes 472

18.3 Decoding for a BCH Code 477

15.1 Exercises 482

Appendix A Rational, Real, and Complex Numbers 485

A.1 Introduction 485

A.2 The Order Relation on the Real Number System 486

A.3 Decimal Representation of Rational Numbers 489

A.4 Complex Numbers 492

A.5 The Polar Form of a Complex Number 495

A.6 Exercises 499

Appendix B Linear Algebra 503

B.1 Vector Spaces 503

B.2 Linear Transformations 513

B.3 Determinants 525

B.4 Eigenvalues and Eigenvectors 536

B.5 Exercises 541

Index 545