Contemporary abstract algebra

Front Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Contents......Page 4
Preface......Page 12
PART 1: Integers and Equivalence Relations......Page 16
Properties of Integers......Page 18
Modular Arithmetic......Page 22
Mathematical Induction......Page 27
Equivalence Relations......Page 30
Functions (Mappings)......Page 33
Exercises......Page 36
Computer Exercises......Page 40
PART 2: Groups......Page 42
Symmetries of a Square......Page 44
The Dihedral Groups......Page 47
Exercises......Page 50
Biography of Niels Abel......Page 54
Definition and Examples of Groups......Page 55
Elementary Properties of Groups......Page 63
Historical Note......Page 66
Exercises......Page 67
Computer Exercises......Page 70
Terminology and Notation......Page 72
Subgroup Tests......Page 73
Examples of Subgroups......Page 76
Exercises......Page 79
Computer Exercises......Page 85
Properties of Cyclic Groups......Page 87
Classification of Subgroups of Cyclic Groups......Page 92
Exercises......Page 96
Computer Exercises......Page 101
Biography of J. J. Sylvester......Page 104
Supplementary Exercises for Chapters 1–4......Page 106
Definition and Notation......Page 110
Cycle Notation......Page 113
Properties of Permutations......Page 115
Exercises......Page 128
Computer Exercises......Page 133
Biography of Augustin Cauchy......Page 136
Definition and Examples......Page 137
Cayley’s Theorem......Page 141
Automorphisms......Page 144
Exercises......Page 148
Computer Exercise......Page 151
Biography of Arthur Cayley......Page 152
Properties of Cosets......Page 153
Lagrange’s Theorem and Consequences......Page 156
An Application of Cosets to Permutation Groups......Page 160
Exercises......Page 164
Computer Exercise......Page 168
Biography of Joseph Lagrange......Page 169
Definition and Examples......Page 170
Properties of External Direct Products......Page 171
The Group of Units Modulo n as an External Direct Product......Page 174
Applications......Page 176
Exercises......Page 182
Computer Exercises......Page 185
Biography of Leonard Adleman......Page 188
Supplementary Exercises for Chapters 5–8......Page 189
Normal Subgroups......Page 193
Factor Groups......Page 195
Applications of Factor Groups......Page 200
Internal Direct Products......Page 203
Exercises......Page 208
Biography of Évariste Galois......Page 214
Definition and Examples......Page 215
Properties of Homomorphisms......Page 217
The First Isomorphism Theorem......Page 221
Exercises......Page 226
Computer Exercise......Page 231
Biography of Camille Jordan......Page 232
The Isomorphism Classes of Abelian Groups......Page 233
Proof of the Fundamental Theorem......Page 238
Exercises......Page 241
Computer Exercises......Page 243
Supplementary Exercises for Chapters 9–11......Page 245
PART 3: Rings......Page 250
Motivation and Definition......Page 252
Examples of Rings......Page 253
Properties of Rings......Page 254
Subrings......Page 255
Exercises......Page 257
Computer Exercises......Page 260
Biography of I. N. Herstein......Page 263
Definition and Examples......Page 264
Fields......Page 265
Characteristic of a Ring......Page 267
Exercises......Page 270
Computer Exercises......Page 274
Biography of Nathan Jacobson......Page 276
Ideals......Page 277
Factor Rings......Page 278
Prime Ideals and Maximal Ideals......Page 282
Exercises......Page 284
Computer Exercises......Page 288
Biography of Richard Dedekind......Page 289
Biography of Emmy Noether......Page 290
Supplementary Exercises for Chapters 12–14......Page 291
Definition and Examples......Page 295
Properties of Ring Homomorphisms......Page 298
The Field of Quotients......Page 300
Exercises......Page 302
Notation and Terminology......Page 308
The Division Algorithm and Consequences......Page 311
Exercises......Page 315
Biography of Saunders Mac Lane......Page 319
Reducibility Tests......Page 320
Irreducibility Tests......Page 323
Unique Factorization in Z[x]......Page 328
Factorization......Page 329
Exercises......Page 331
Computer Exercises......Page 334
Biography of Serge Lang......Page 336
Irreducibles, Primes......Page 337
Historical Discussion of Fermat’s Last Theorem......Page 340
Unique Factorization Domains......Page 343
Euclidean Domains......Page 346
Exercises......Page 350
Computer Exercise......Page 352
Biography of Sophie Germain......Page 354
Biography of Andrew Wiles......Page 355
Supplementary Exercises for Chapters 15–18......Page 356
PART 4: Fields......Page 358
Definition and Examples......Page 360
Subspaces......Page 361
Linear Independence......Page 362
Exercises......Page 364
Biography of Emil Artin......Page 367
Biography of Olga Taussky-Todd......Page 368
The Fundamental Theorem of Field Theory......Page 369
Splitting Fields......Page 371
Zeros of an Irreducible Polynomial......Page 377
Exercises......Page 381
Biography of Leopold Kronecker......Page 384
Characterization of Extensions......Page 385
Finite Extensions......Page 387
Properties of Algebraic Extensions......Page 391
Exercises......Page 393
Biography of Irving Kaplansky......Page 396
Classification of Finite Fields......Page 397
Structure of Finite Fields......Page 398
Subfields of a Finite Field......Page 402
Exercises......Page 404
Computer Exercises......Page 406
Biography of L. E. Dickson......Page 407
Historical Discussion of Geometric Constructions......Page 408
Constructible Numbers......Page 409
Exercises......Page 411
Supplementary Exercises for Chapters 19–23......Page 414
PART 5: Special Topics......Page 416
Conjugacy Classes......Page 418
The Class Equation......Page 419
The Probability That Two Elements Commute......Page 420
The Sylow Theorems......Page 421
Applications of Sylow Theorems......Page 426
Computer Exercise......Page 433
Biography of Ludwig Sylow......Page 434
Historical Background......Page 435
Nonsimplicity Tests......Page 440
The Simplicity of A5......Page 444
The Cole Prize......Page 445
Exercises......Page 446
Computer Exercises......Page 447
Biography of Michael Aschbacher......Page 449
Biography of Daniel Gorenstein......Page 450
Biography of John Thompson......Page 451
Motivation......Page 452
Definitions and Notation......Page 453
Free Group......Page 454
Generators and Relations......Page 455
Classification of Groups of Order Up to 15......Page 459
Characterization of Dihedral Groups......Page 461
Realizing the Dihedral Groups with Mirrors......Page 462
Exercises......Page 464
Biography of Marshall Hall, Jr.......Page 467
Isometries......Page 468
Classification of Finite Plane Symmetry Groups......Page 470
Classification of Finite Groups of Rotations in R3......Page 471
Exercises......Page 473
The Frieze Groups......Page 476
The Crystallographic Groups......Page 482
Identification of Plane Periodic Patterns......Page 488
Exercises......Page 494
Biography of M. C. Escher......Page 499
Biography of George Pólya......Page 500
Biography of John H. Conway......Page 501
Motivation......Page 502
Burnside’s Theorem......Page 503
Applications......Page 505
Group Action......Page 508
Exercises......Page 509
Biography of William Burnside......Page 512
The Cayley Digraph of a Group......Page 513
Hamiltonian Circuits and Paths......Page 517
Some Applications......Page 523
Exercises......Page 526
Biography of William Rowan Hamilton......Page 531
Biography of Paul Erdös......Page 532
Motivation......Page 533
Linear Codes......Page 538
Parity-Check Matrix Decoding......Page 543
Coset Decoding......Page 546
Historical Note: The Ubiquitous Reed-Solomon Codes......Page 550
Exercises......Page 552
Biography of Richard W. Hamming......Page 557
Biography of Jessie MacWilliams......Page 558
Biography of Vera Pless......Page 559
Fundamental Theorem of Galois Theory......Page 560
Solvability of Polynomials by Radicals......Page 567
Insolvability of a Quintic......Page 571
Exercises......Page 572
Biography of Philip Hall......Page 575
Motivation......Page 576
Cyclotomic Polynomials......Page 577
The Constructible Regular n-gons......Page 581
Exercises......Page 583
Computer Exercise......Page 584
Biography of Carl Friedrich Gauss......Page 585
Supplementary Exercises for Chapters 24–33......Page 586
Selected Answers......Page 590
Text Credits......Page 629
Photo Credits......Page 631
Index of Mathematicians......Page 632
Index of Terms......Page 634