CONTEMPORARY ABSTRACT ALGEBRA, EIGHTH EDITION provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students.
Author(s): Joseph Gallian
Edition: 8
Publisher: Cengage Learning
Year: 2012
Language: English
Pages: 664
Cover
......Page 1
Title Page
......Page 5
Copyright
......Page 6
Contents ......Page 10
Preface ......Page 17
Part 1: Integers and Equivalence Relations ......Page 21
Properties of Integers ......Page 23
Modular Arithmetic ......Page 26
Complex Numbers ......Page 33
Mathematical Induction ......Page 34
Equivalence Relations ......Page 37
Functions (Mappings) ......Page 40
Exercises ......Page 43
Suggested Readings......Page 47
Part 2: Groups ......Page 49
Symmetries of a Square ......Page 51
The Dihedral Groups ......Page 54
Exercises ......Page 57
Suggested Website......Page 60
Niels Abel......Page 61
Definition and Examples of Groups ......Page 62
Elementary Properties of Groups ......Page 70
Historical Note ......Page 73
Exercises ......Page 74
Computer Exercises......Page 78
Suggested Readings......Page 79
Terminology and Notation ......Page 80
Subgroup Tests ......Page 81
Examples of Subgroups ......Page 85
Exercises ......Page 88
Suggested Readings......Page 95
Suggested Software......Page 96
Properties of Cyclic Groups ......Page 97
Classification of Subgroups of Cyclic Groups ......Page 102
Exercises ......Page 107
Suggested Reading......Page 112
James Joseph Sylvester......Page 113
Supplementary Exercises for Chapters 1-4 ......Page 115
Definition and Notation ......Page 119
Cycle Notation ......Page 122
Properties of Permutations ......Page 124
A Check-Digit Scheme Based on D5 ......Page 135
Exercises ......Page 138
Suggested Readings......Page 144
Suggested Software......Page 145
Augustin Cauchy......Page 146
Definition and Examples......Page 147
Cayley's Theorem ......Page 151
Properties of Isomorphisms ......Page 153
Automorphisms ......Page 154
Exercises ......Page 158
Computer Exercises......Page 162
Arthur Cayley......Page 163
Properties of Cosets ......Page 164
Lagrange's Theorem and Consequences ......Page 167
An Application of Cosets to Permutation Groups ......Page 171
The Rotation Group of a Cube and a Soccer Ball ......Page 173
An Application of Cosets to the Rubik's Cube ......Page 175
Exercises ......Page 176
Computer Exercises......Page 180
Joseph Lagrange......Page 181
Definition and Examples ......Page 182
Properties of External Direct Products ......Page 183
The Group of Units Modulo n as an External Direct Product ......Page 186
Applications ......Page 188
Exercises ......Page 194
Suggested Readings......Page 198
Leonard Adleman......Page 200
Supplementary Exercises for Chapters 5-8 ......Page 201
Normal Subgroups ......Page 205
Factor Groups ......Page 207
Applications of Factor Groups ......Page 213
Internal Direct Products ......Page 215
Exercises ......Page 220
Suggested Readings......Page 225
Evariste Galois......Page 227
Definition and Examples ......Page 228
Properties of Homomorphisms ......Page 230
The First Isomorphism Theorem ......Page 234
Exercises ......Page 239
Computer Exercise......Page 244
Camille Jordan......Page 245
The Isomorphism Classes of Abelian Groups ......Page 246
Proof of the Fundamental Theorem ......Page 251
Exercises ......Page 254
Computer Exercises......Page 256
Suggested Website......Page 257
Supplementary Exercises for Chapters 9-11 ......Page 258
Part 3: Rings ......Page 263
Motivation and Definition ......Page 265
Examples of Rings ......Page 266
Properties of Rings ......Page 267
Subrings ......Page 268
Exercises ......Page 270
Suggested Reading......Page 273
I. N. Herstein......Page 274
Definition and Examples ......Page 275
Fields ......Page 276
Characteristic of a Ring ......Page 278
Exercises ......Page 281
Suggested Readings......Page 285
Nathan Jacobson......Page 286
Ideals ......Page 287
Factor Rings ......Page 288
Prime Ideals and Maximal Ideals ......Page 292
Exercises ......Page 294
Computer Exercises......Page 298
Richard Dedekind......Page 299
Emmy Noether......Page 300
Supplementary Exercises for Chapters 12-14 ......Page 301
Definition and Examples ......Page 305
Properties of Ring Homomorphisms ......Page 308
The Field of Quotients ......Page 310
Exercises ......Page 312
Suggested Readings......Page 316
Suggested Website......Page 317
Notation and Terminology ......Page 318
The Division Algorithm and Consequences ......Page 321
Exercises ......Page 325
Saunders Mac Lane......Page 330
Reducibility Tests ......Page 331
Irreducibility Tests ......Page 334
Unique Factorization in Z[x] ......Page 339
Weird Dice: An Application of Unique Factorization ......Page 340
Exercises ......Page 342
Computer Exercises......Page 345
Suggested Readings......Page 346
Serge Lang......Page 347
Irreducibles, Primes......Page 348
Historical Discussion of Fermat's Last Theorem......Page 351
Unique Factorization Domains ......Page 354
Euclidean Domains ......Page 357
Exercises ......Page 361
References......Page 363
Suggested Websites......Page 364
Sophie Germain......Page 365
Andrew Wiles......Page 366
Supplementary Exercises for Chapters 15-18 ......Page 367
Part 4: Fields ......Page 369
Definition and Examples ......Page 371
Subspaces ......Page 372
Linear Independence ......Page 373
Exercises ......Page 375
Emil Artin......Page 378
Olga Taussky-Todd......Page 379
The Fundamental Theorem of Field Theory ......Page 380
Splitting Fields ......Page 382
Zeros of an Irreducible Polynomial ......Page 388
Exercises ......Page 392
Leopold Kronecker......Page 395
Characterization of Extensions ......Page 396
Finite Extensions ......Page 398
Properties of Algebraic Extensions ......Page 402
Exercises ......Page 404
Suggested Readings......Page 406
Irving Kaplansky......Page 407
Classification of Finite Fields ......Page 408
Structure of Finite Fields ......Page 409
Subfields of a Finite Field ......Page 413
Exercises ......Page 415
Suggested Reading......Page 417
L. E. Dickson......Page 418
Historical Discussion of Geometric Constructions ......Page 419
Constructible Numbers ......Page 420
Exercises ......Page 422
Suggested Website......Page 424
Supplementary Exercises for Chapters 19-23 ......Page 425
Part 5: Special Topics ......Page 427
Conjugacy Classes ......Page 429
The Class Equation ......Page 430
The Probability that Two Elements Commute......Page 431
The Sylow Theorems ......Page 432
Applications of Sylow Theorems ......Page 437
Exercises ......Page 441
Computer Exercises......Page 445
Suggested Readings......Page 446
Ludwig Sylow......Page 447
Historical Background ......Page 448
Nonsimplicity Tests ......Page 453
The Simplicity of A5 ......Page 457
The Cole Prize ......Page 458
Exercises ......Page 459
References......Page 460
Suggested Readings......Page 461
Michael Aschbacher......Page 462
Daniel Gorenstein......Page 463
John Thompson......Page 464
Motivation ......Page 465
Definitions and Notation ......Page 466
Free Group ......Page 467
Generators and Relations ......Page 468
Classification of Groups of Order up to 15......Page 472
Characterization of Dihedral Groups ......Page 474
Realizing the Dihedral Groups with Mirrors ......Page 475
Exercises ......Page 477
References......Page 478
Suggested Readings......Page 479
Marshall Hall, Jr.......Page 480
Isometries ......Page 481
Classification of Finite Plane Symmetry Groups ......Page 483
Classification of Finite Groups of Rotations in R3 ......Page 484
Exercises ......Page 486
Suggested Website......Page 488
The Frieze Groups ......Page 489
The Crystallographic Groups ......Page 495
Identification of Plane Periodic Patterns ......Page 501
Exercises ......Page 507
References......Page 509
Suggested Readings......Page 510
Suggested Websites......Page 511
M. C. Escher......Page 512
George Polya......Page 513
John H. Conway......Page 514
Motivation ......Page 515
Burnside's Theorem ......Page 516
Applications ......Page 518
Group Action ......Page 521
Exercises ......Page 522
Suggested Readings......Page 524
William Burnside......Page 525
The Cayley Digraph of a Group ......Page 526
Hamiltonian Circuits and Paths ......Page 530
Some Applications ......Page 536
Exercises ......Page 539
Suggested Readings......Page 542
Suggested Software......Page 543
William Rowan Hamilton......Page 544
Paul Erdos......Page 545
Motivation ......Page 546
Linear Codes ......Page 551
Parity-Check Matrix Decoding ......Page 556
Coset Decoding ......Page 559
Historical Note: The Ubiquitous Reed-Solomon Codes ......Page 563
Exercises ......Page 565
Suggested Readings......Page 569
Richard W. Hamming......Page 570
Jessie MacWilliams......Page 571
Vera Pless......Page 572
Fundamental Theorem of Galois Theory ......Page 573
Solvability of Polynomials by Radicals ......Page 580
Insolvability of a Quintic ......Page 584
Exercises ......Page 585
Suggested Website......Page 588
Philip Hall......Page 589
Motivation ......Page 590
Cyclotomic Polynomials ......Page 591
The Constructible Regular n-gons ......Page 595
Exercises ......Page 597
Computer Exercises......Page 598
Carl Friedrich Gauss......Page 599
Manjul Bhargava......Page 600
Supplementary Exercises for Chapters 24-33 ......Page 601
Selected Answers......Page 605
Index of Mathematicians......Page 649
Index of Terms......Page 651
