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Contemporary Abstract Algebra

by: Gallian, Joseph A.

Contemporary Abstract Algebra

by: Gallian, Joseph A.

ISBN 13:
9780395861790
ISBN 10:
0395861799
Edition: 4th
Format: Paperback
Copyright: 09/29/1997
Publisher: Brooks Cole

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Summary

Joseph Gallian is a well-known active researcher and award-winning teacher. His Contemporary Abstract Algebra, 6/e, includes challenging topics in abstract algebra as well as numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings that give the subject a current feel and makes the content interesting and relevant for students.

Preface

PART 1 Integers and Equivalence Relations

(26)

CHAPTER 0 Preliminaries

(24)

Properties of Integers

(5)

Modular Arithmetic

(5)

Mathematical Induction

(3)

Equivalence Relations

(2)

Functions (Mappings)

(3)

Exercises

(3)

Computer Exercises

(3)

PART 2 Groups

(196)

CHAPTER 1 Introduction to Groups

(11)

Symmetries of a Square

(3)

The Dihedral Groups

(3)

Exercises

(4)

Biography of Niels Abel

(1)

CHAPTER 2 Groups

(16)

Definition and Examples of Groups

(8)

Elementary Properties of Groups

(2)

Historical Note

(1)

Exercises

(4)

Computer Exercises

(2)

CHAPTER 3 Finite Groups; Subgroups

(15)

Terminology and Notation

(2)

Subgroup Tests

(2)

Examples of Subgroups

(5)

Exercises

(4)

Computer Exercises

(2)

CHAPTER 4 Cyclic Groups

(15)

Properties of Cyclic Groups

(4)

Classification of Subgroups of Cyclic Groups

(4)

Exercises

(3)

Computer Excercises

(2)

Biography of J.J. Sylvester

(2)

Supplementary Exercises for Chapters 1-4

(4)

CHAPTER 5 Permutation Groups

(25)

Definition and Notation

(3)

Cycle Notation

(2)

Properties of Permutations

(9)

A Check-Digit Scheme Based on D(5)

104

(3)

Exercises

107

(3)

Computer Exercises

110

(3)

Biography of Augustin Cauchy

113

(2)

CHAPTER 6 Isomorphisms

115

(17)

Motivation

115

(1)

Definition and Examples

116

(3)

Cayley's Theorem

119

(2)

Properties of Isomorphisms

121

(1)

Automorphisms

122

(4)

Exercises

126

(4)

Biography of Arthur Cayley

130

(2)

CHAPTER 7 Cosets and Lagrange's Theorem

132

(17)

Properties of Cosets

132

(3)

Lagrange's Theorem and Consequences

135

(2)

An Application of Cosets to Permutation Groups

137

(2)

The Rotation Group of a Cube and a Soccer Ball

139

(3)

Exercises

142

(5)

Biography of Joseph Lagrange

147

(2)

CHAPTER 8 External Direct Products

149

(19)

Definition and Examples

149

(2)

Properties of External Direct Products

151

(3)

The Group of Units Modulo n as an External Direct Product

154

(2)

Applications

156

(5)

Exercises

161

(3)

Computer Exercises

164

(2)

Biography of Leonard Adleman

166

(2)

Supplementary Exercises for Chapters 5-8

168

(3)

CHAPTER 9 Normal Subgroups and Factor Groups

171

(21)

Normal Subgroups

171

(1)

Factor Groups

172

(6)

Applications of Factor Groups

178

(3)

Internal Direct Products

181

(4)

Exercises

185

(5)

Biography of Evariste Galois

190

(2)

CHAPTER 10 Group Homomorphisms

192

(17)

Definition and Examples

192

(2)

Properties of Homomorphisms

194

(4)

The First Isomorphism Theorem

198

(5)

Exercises

203

(4)

Biography of Camille Jordan

207

(2)

CHAPTER 11 Fundamental Theorem of Finite Abelian Groups

209

(12)

The Fundamental Theorem

209

(1)

The Isomorphism Classes of Abelian Groups

210

(4)

Proof of the Fundamental Theorem

214

(3)

Exercises

217

(2)

Computer Exercises

219

(2)

Supplementary Exercises for Chapters 9-11

221

(2)

PART 3 Rings

223

(106)

CHAPTER 12 Introduction to Rings

225

(11)

Motivation and Definition

225

(1)

Examples of Rings

226

(1)

Properties of Rings

227

(1)

Subrings

228

(5)

Exercises

233

(1)

Computer Exercises

233

(2)

Biography of I.N. Herstein

235

(1)

CHAPTER 13 Integral Domains

236

(12)

Definition and Examples

236

(2)

Fields

238

(2)

Characteristic of a Ring

240

(2)

Exercises

242

(4)

Computer Exercises

246

(1)

Biography of Nathan Jacobson

247

(1)

CHAPTER 14 Ideals and Factor Rings

248

(15)

Ideals

248

(1)

Factor Rings

249

(4)

Prime Ideals and Maximal Ideals

253

(2)

Exercises

255

(4)

Biography of Richard Dedekind

259

(2)

Biography of Emmy Noether

261

(2)

Supplementary Exercises for Chapters 12-14

263

(3)

CHAPTER 15 Ring Homomorphisms

266

(13)

Definition and Examples

266

(3)

Properties of Ring Homomorphisms

269

(3)

The Field of Quotients

272

(2)

Exercises

274

(5)

CHAPTER 16 Polynomial Rings

279

(12)

Notation and Terminology

279

(3)

The Division Algorithm and Consequences

282

(4)

Exercises

286

(4)

Biography of Saunders Mac Lane

290

(1)

CHAPTER 17 Factorization of Polynomials

291

(17)

Reducibility Tests

291

(3)

Irreducibility Tests

294

(5)

Unique Factorization in Z[x]

299

(2)

Weird Dice: An Application of Unique Factorization

301

(2)

Exercises

303

(3)

Computer Exercises

306

(2)

CHAPTER 18 Divisibility in Integral Domains

308

(19)

Irreducibles, Primes

308

(3)

Historical Discussion of Fermat's Last Theorem

311

(3)

Unique Factorization Domains

314

(3)

Euclidean Domains

317

(3)

Exercises

320

(4)

Biography of Sophie Germain

324

(2)

Biography of Andrew Wiles

326

(1)

Supplementary Exercises for Chapters 15-18

327

(2)

PART 4 Fields

329

(64)

CHAPTER 19 Vector Spaces

331

(10)

Definition and Examples

331

(1)

Subspaces

332

(1)

Linear Independence

333

(2)

Exercises

335

(3)

Biography of Emil Artin

338

(2)

Biography of Olga Taussky-Todd

340

(1)

CHAPTER 20 Extension Fields

341

(18)

The Fundamental Theorem of Field Theory

341

(2)

Splitting Fields

343

(7)

Zeros of an Irreducible Polynomial

350

(4)

Exercises

354

(3)

Biography of Leopold Kronecker

357

(2)

CHAPTER 21 Algebraic Extensions

359

(13)

Characterization of Extensions

359

(2)

Finite Extensions

361

(4)

Properties of Algebraic Extensions

365

(2)

Exercises

367

(3)

Biography of Irving Kaplansky

370

(2)

CHAPTER 22 Finite Fields

372

(11)

Classification of Finite Fields

372

(1)

Structure of Finite Fields

373

(4)

Subfields of a Finite Field

377

(2)

Exercises

379

(2)

Biography of L.E. Dickson

381

(2)

CHAPTER 23 Geometric Constructions

383

(7)

Historical Discussion of Geometric Constructions

383

(1)

Constructible Numbers

384

(2)

Angle-Trisectors and Circle-Squarers

386

(1)

Exercises

387

(3)

Supplementary Exercises for Chapter 19-23

390

(3)

PART 5 Special Topics

393

CHAPTER 24 Sylow Theorems

395

(18)

Conjugacy Classes

395

(1)

The Class Equation

396

(1)

The Probability That Two Elements Commute

397

(1)

The Sylow Theorems

398

(5)

Applications of Sylow Theorems

403

(4)

Exercises

407

(5)

Biography of Ludvig Sylow

412

(1)

CHAPTER 25 Finite Simple Groups

413

(21)

Historical Background

413

(4)

Nonsimplicity Tests

417

(6)

The Simplicity of A(5)

423

(1)

The Fields Medal

424

(1)

The Cole Prize

424

(1)

Exercises

424

(2)

Computer Exercises

426

(2)

Biography of Michael Aschbacher

428

(2)

Biography of Daniel Gorenstein

430

(2)

Biography of John Thompson

432

(2)

CHAPTER 26 Generators and Relations

434

(19)

Motivation

434

(1)

Definitions and Notation

435

(1)

Free Group

436

(1)

Generators and Relations

437

(4)

Classification of Groups of Order up to 15

441

(2)

Characterization of Dihedral Groups

443

(2)

Realizing the Dihedral Groups with Mirrors

445

(2)

Exercises

447

(4)

Biography of Marshall Hall, Jr.

451

(2)

CHAPTER 27 Symmetry Groups

453

(9)

Isometries

453

(2)

Classification of Finite Plane Symmetry Groups

455

(2)

Classification of Finite Groups of Rotations in R(3)

457

(2)

Exercises

459

(3)

CHAPTER 28 Frieze Groups and Crystallographic Groups

462

(28)

The Frieze Groups

462

(5)

The Crystallographic Groups

467

(2)

Identification of Plane Periodic Patterns

469

(11)

Exercises

480

(5)

Biography of M.C. Escher

485

(2)

Biography of George Polya

487

(1)

Biography of John H. Conway

488

(2)

CHAPTER 29 Symmetry and Counting

490

(12)

Motivation

490

(2)

Burnside's Theorem

492

(1)

Applications

493

(4)

Group Action

497

(1)

Exercises

498

(2)

Biography of William Burnside

500

(2)

CHAPTER 30 Cayley Digraphs of Groups

502

(23)

Motivation

502

(1)

The Cayley Digraphs of a Groups

502

(4)

Hamiltonian Circuits and Paths

506

(7)

Some Applications

513

(2)

Exercises

515

(6)

Biography of William Rowan Hamilton

521

(2)

Biography of Paul Erdos

523

(2)

CHAPTER 31 Introduction to Algebraic Coding Theory

525

(29)

Motivation

525

(5)

Linear Codes

530

(5)

Parity-Check Matrix Decoding

535

(3)

Coset Decoding

538

(4)

Historical Note: Reed-Solomon Codes

542

(2)

Exercises

544

(5)

Biography of Richard W. Hamming

549

(2)

Biography of Jessie MacWilliams

551

(1)

Biography of Vera Pless

552

(2)

CHAPTER 32 An Introduction to Galois Theory

554

(17)

Fundamental Theorem of Galois Theory

554

(7)

Solvability of Polynomials by Radicals

561

(5)

Insolvability of a Quintic

566

(1)

Exercises

567

(3)

Biography of Philip Hall

570

(1)

CHAPTER 33 Cyclotomic Extensions

571

(11)

Motivation

571

(1)

Cyclotomic Polynomials

572

(4)

The Constructible Regular n-gons

576

(2)

Exercises

578

(1)

Computer Exercise

579

(1)

Biography of Carl Friedrich Gauss

580

(2)

Supplementary Exercises for Chapters 24-33

582

Selected Answers

(38)

Text Credits

A39

(2)

Photo Credits

A41

(2)

Notations

A43

(4)

Index of Mathematicians

A47

(2)

Index of Terms

A49

Amazon no longer offers textbook rentals. We do!

Contemporary Abstract Algebra

Contemporary Abstract Algebra

9780395861790

0395861799

Summary

Table of Contents

Supplemental Materials

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