Contemporary Abstract Algebra / Edition 6

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Overview

The seventh edition of Contemporary Abstract Algebra, by Joseph A. Gallian, Provides a solid introduction to the traditional topics in abstract algebra while conveying that it is a contemporary subject used daily by working mathematicians, computer scientist, and chemists. The text includes numerous theoretical and computational exercises, figures, and tables to teach you how to work out problems, as well as to write proofs. Additionally, the author provides biographies, poems, song Lyrics, historical notes, and much more to make reading the text an interesting, accessible and enjoyable experience. Contemporary Abstract Algebra will keep you engaged and gives you a great introduction to an important subject.
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Product Details

  • ISBN-13: 9780618514717
  • Publisher: Cengage Learning
  • Publication date: 12/15/2004
  • Edition description: 6TH
  • Edition number: 6
  • Pages: 640
  • Product dimensions: 6.72 (w) x 9.28 (h) x 1.12 (d)

Meet the Author

Joseph Gallian earned his PhD from Notre Dame. In addition to receiving numerous awards for his teaching and exposition, he served, first, as the Second Vice President, and, then, as the President of the MAA. He has served on 40 national committees, chairing ten of them. He has published over 100 articles and authored six books. Numerous articles about his work have appeared in the national news outlets, including the New York Times, the Washington Post, the Boston Globe, and Newsweek, among many others.

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Table of Contents

Preface xi

Part 1 Integers and Equivalence Relations 1

0 Preliminaries 3

Properties of Integers 3

Madular Arithmetic 7

Mathematical Induction 12

Equivalence Relations 15

Functions (Mappings) 18

Exercises 21

Computer Exercises 25

Part 2 Groups 27

1 Introduction to Groups 29

Symmetries of a Square 29

The Dihedral Groups 32

Exercises 35

Biography of Niels Abel 39

2 Groups 40

Definition and Examples of Groups|o40

Elementary Properties of Groups 48

Historical Note 51

Exercises 52

Computer Exercises 55

3 Finite Groups; Subgroups 57

Terminology and Notation 57

Subgroup Tests 58

Examples of Subgroups 61

Exercises 64

Computer Exercises 70

4 Cyclic Groups 72

Properties of Cycle Groups 72

Classification of Subgroups of Cyclic Groups 77

Exercises 81

Computer Exercises 86

Biography of J. J. Sylvester 89

Supplementary Exercises for Chapters1-4 91

5 Permutation Groups 95

Difinition and Notation 95

Cycle Nation 98

Properties of Permutations 100

A Check Digit Scheme Based on D5 110

Exercises 113

Computer Exercises 118

Biography of Augustin Cauchy 121

6 Isomorphisms 122

Motivation 122

Dfinition and Examples 122

Cayley's Theorem 126

Properties of Isomorphisms 128

Automorphisms 129

Exercises 133

Computer Exercise 136

Biography of Arthur Cayley 137

7 Cosets and Lagrange's Theorem 138

Properties of Cosets 138

Lagrange's Theorem and Consequences 141

An Application of Cosets of Permutation Groups 145

The Rotation Group of a Cube and a Soccer Ball 146

Exercises 149

Computer Exercise 153

Biography of Joseph Lagrange 154

8 External Direct Products 155

Definition and Examples 155

Properties ofExternal Direct Products 156

The Group of Units Modulo n as an External Direct Products 159

Applications 161

Exercises 167

Computer Exercises 170

Biorgaphy of Leonard Adleman 173

Supplementary Exercises for Chapters 5-8 174

9 Normal Subgroups and Factor Groups 178

Normal Subgroups 178

Factor Groups 180

Applicatons of Factor Groups 185

Internal Direct Products 188

Exercises 193

Biography of Evariste Galois 199

10 Group Homomorphisms 200

Difinition and Examples 200

Properties Of Homomorphisms 202

The First Isomorphism Theorem 206

Exercises 211

Computer Exercise 216

Biography of Camille Jordan 217

11 Fundamental Theorem of Finite Abelian Groups 218

The Fundamental Theorem 218

The Isomorphism Classes of Abelian Groups 218

Proof of the Fundamental Theorem 223

Exercises 226

Computer Exercises 228

Supplementary Exercises for Chapter 9-11 230

Part 3 Rings 235

12 Introduction to Rings 237

Motivation and Definition 237

Examples of Rings 238

Properties of Rings 239

Subrings 240

Exercises 242

Computer Exercises 245

Biography of I. N. Herstein 248

13 Integral Domains 249

Definition and Examples 249

Fields 250

Characteristic of a Ring 225

Exercises 255

Computer Exercises 259

Biography of Nathan Jacobson 261

14 Ideals and Factor Rings 262

Ideals 262

Factor Rings 263

Prime Ideals and Maximal Ideals 267

Exercises 269

Computer Exercises 273

Biography of Richard Dedekind 274

Biography of Emmy Noether 275

Supplementary Exercises for Chapters 12-14 276

15 Ring Homomorphisms 280

Definition and Example 280

Properties of Ring Homomorphisms 283

The Field of Quotients 285

Exercises 287

16 Polynomial Rings 293

Notation and Terminology 293

The Division Algorithm and Consequences 296

Exercises 300

Biography of Sounders Mac Lane 304

17 Factorization of Polynomials 305

Reducibility Tests 305

Irreducibility Tests 308

Unique Factorization in Z[x] 313

Weird Dice: An Application of Unique Factorization 314

Exercises 316

Computer Exercises 319

Biography of Serge Lang 321

18 Divisibility in Integral Domains 322

Irreducibles, Primes 322

Historical Discussion of Fermat's Last Theorem 325

Unique Factorization Domains 328

Euclidean Domains 331

Exercises 335

Comupter Exercise 337

Biography of Sophie Germain 339

Biography of Andrew Wiles 340

Supplementary Exercises for Chapters 15-18 341

Part 4 Fields 343

19 Vector Spaces 345

Definition and Examples 345

Subspaces 346

Linear Independence 347

Exercises 349

Biography of Emil Artin 352

Biography of Olga Taussky-Todd 353

20 Extension Fields 354

The Fundamental Theorem of Field theory 354

Splitting Fields 356

Zeros of an Irreducible Polynomial 362

Exercises 366

Biography of Leopold Kronecker 369

21 Algebraci Extensions 370

Characterization of Extensions 370

Finite Extensions 372

Properties of Algebraic Extensions 376

Exercises 378

Biography of Irving Kaplansky 381

22 Finite Fields 382

Classification of Finite Fields 382

Struction of Finite Fields 383

Subfields of a Finite Field 387

Exercises 389

Computer Exercises 391

Biography of L. E. Dickson 392

23 Geometric Constructions 393

Historical Discussion of Geometric Constructions 393

Constructible Numbers 394

Angle-Trisectors and Circle-Squarers 396

Exercises 396

Supplementary Exercises for Chapters 19-23 399

Part 5 Special Topics 401

24 Sylow Theorems 403

Conjugacy Classes 403

The Class Equation 404

The Probability That Two Elements Commute 405

The Sylow Theorems 406

Applications of Sylow Theorems 411

Exercises 414

Computer Exercise 418

Biography of Ludwig Sylow 419

25 Finite Simple Groups 420

Historical Background 420

Nonsimplicity Tests 245

The Simplicity of A5 429

The Fields Medal 430

The Cole Prize 430

Execises 431

Computer Exercises 432

Biography of Michael Aschbacher 434

Biography of Daniel Gorenstein 435

Biography of John Thompson 436

26 Generators and Relations 437

Motivation 437

Definitions and Notation 438

Free Group 439

Generators and Relations 440

Classification of Groups of Order Up to 15 444

Characterization of Dihedral Group 446

Realizing the Dihedral Groups with Mirrors 447

Exercises 449

Biography of Marshall Hall, Jr. 452

27 Symmetry Groups 453

Isometries 453

Classification of Finite Plane Symmetry Group 455

Classification of Finite Groups of Rotations in R3 456

Exercises 458

28 Frieze Groups and Crystallographic Groups 461

The Frieze Groups 461

The Crystallographic Groups 467

Identification of Plane Periodic Patterns 473

Exercises 479

Biography of M. C. Escher 484

Biography of George Polya 485

Biography of John H. Conway 486

29 Symmetry and Counting 487

Motivation 487

Burnside's Theorem 488

Applications 490

Group Action 493

Exercises 494

Biography of William Burnside 497

30 Cayley Digraphs of Groups 498

Motivaton 498

The Cayley Digraph of a Group 498

Hamiltonian Circuits and Paths 502

Some Apllications 508

Exercises 511

Biography of William Rowan Hamilton 516

Biography of Paul Erdos 517

31 Indtoduction to Algebraic Coding Theory 518

Motivation 518

Liner Codes 523

Parity-Check Matrix Decoding 528

Coset Decoding 531

Hestorical Note: The Ubiquitous Reed-Solomon Codes 535

Exercises 537

Biography of Richard W. Hamming 542

Biography of Jessie Mac Williams 543

Biography of Vera Pless 544

32 An Introduction to Galois Theory 545

Fundamental Theorem of Galois Theory 545

Solvability of Polynomials by Radicals 552

Insolvability of a Quintic 556

Exercises 557

Biography of Philip Hall 560

33 Cyclotomic Extensions 561

Motivation 561

Cyclotomic Polynomials 562

The Constructible Regular n-Gons 566

Exercises 568

Computer Exercis 569

Biography of Carl Friedrich Gauss 570

Biography of Manjul Bhargava 571

Supplementary Exercises for Chapters 24-33 572

Selected Answers A1

Text Credits A40

Photo Credits A42

Index of Mathematicians A43

Index of Terms A45

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