Contemporary abstract algebra

Front cover......Page 1
Title page......Page 3
Date-line......Page 4
Acknowledgments......Page 5
Preface to the 2nd edition......Page 9
Preface to the 1st edition......Page 11
Contents......Page 15
Integers and Equivalence Relations......Page 27
Properties of Integers......Page 29
Modular Arithmetic......Page 33
Mathematical Induction......Page 36
Equivalence Relations......Page 39
Functions (Mappings)......Page 41
Exercises......Page 43
Groups......Page 47
Symmetries of a Square......Page 49
The Dihedral Groups......Page 52
Exercises......Page 54
Biography of Niels Abel......Page 58
Definition and Examples of Groups......Page 59
Elementary Properties of Groups......Page 66
Applications of Modular Arithmetic......Page 68
Historical Note......Page 71
Exercises......Page 72
Programming Exercises......Page 76
Terminology and Notation......Page 79
Subgroup Tests......Page 80
Examples of Subgroups......Page 82
Exercises......Page 86
Programming Exercises......Page 90
Properties of Cyclic Groups......Page 92
Classification of Subgroups of Cyclic Groups......Page 96
Exercises......Page 98
Programming Exercises......Page 102
Biography of J. J. Sylvester......Page 104
Supplementary Exercises for Chapters 1-4......Page 106
Definition and Notation......Page 109
Cycle Notation......Page 112
Properties of Permutations......Page 114
A Check-Digit Scheme Based on $D_5$......Page 118
Exercises......Page 120
Programming Exercise......Page 122
Biography of Augustin Cauchy......Page 124
Motivation......Page 125
Definition and Examples......Page 126
Cayley\'s Theorem......Page 129
Properties of Isomorphisms......Page 130
Automorphisms......Page 131
Exercises......Page 134
Biography of Arthur Cayley......Page 138
Definition and Examples......Page 139
Properties of External Direct Products......Page 140
Exercises......Page 142
Programming Exercises......Page 144
Definition and Examples......Page 145
The Group of Units Modulo $n$ As an Internal and External Direct Product......Page 148
Exercises......Page 150
Programming Exercises......Page 152
Supplementary Exercises for Chapters 5-8......Page 154
Properties of Cosets......Page 156
Lagrange\'s Theorem and Consequences......Page 158
An Application of Cosets to Permutation Groups......Page 161
The Rotation Group of a Cube and a Soccer Ball......Page 162
Exercises......Page 165
Biography of Joseph Lagrange......Page 170
Normal Subgroups......Page 171
Factor Groups......Page 172
Applications of Factor Groups......Page 177
Exercises......Page 179
Biography of Evariste Galois......Page 184
Definition and Examples......Page 185
Properties of Homomorphisms......Page 187
The First Isomorphism Theorem......Page 188
Exercises......Page 193
Biography of Camille Jordan......Page 197
The Fundamental Theorem......Page 198
The Isomorphism Classes of Abelian Groups......Page 199
Proof of the Fundamental Theorem......Page 203
Exercises......Page 205
Programming Exercises......Page 207
Supplementary Exercises for Chapters 9-12......Page 209
Rings......Page 211
Motivation and Definition......Page 213
Examples of Rings......Page 214
Exercises......Page 215
Programming Exercise......Page 217
Properties of Rings......Page 218
Subrings......Page 219
Exercises......Page 221
Programming Exercise......Page 223
Biography of I. N. Herstein......Page 224
Definition and Examples......Page 225
Fields......Page 227
Characteristic of a Ring......Page 228
Exercises......Page 230
Programming Exercises......Page 234
Biography of Nathan Jacobson......Page 235
Ideals......Page 236
Factor Rings......Page 237
Prime Ideals and Maximal Ideals......Page 239
Exercises......Page 241
Biography of Richard Dedekind......Page 244
Biography of Emmy Noether......Page 245
Supplementary Exercises for Chapters 13-16......Page 246
Definition and Examples......Page 248
Properties of Ring Homomorphisms......Page 250
The Field of Quotients......Page 253
Exercises......Page 254
Notation and Terminology......Page 259
The Division Algorithm and Consequences......Page 262
Exercises......Page 265
Reducibility Tests......Page 268
Irreducibility Tests......Page 270
Unique Factorization in $Z[x]$......Page 275
Weird Dice: An Application of Unique Factorization......Page 276
Exercises......Page 278
Programming Exercises......Page 281
Biography of Carl Friedrich Gauss......Page 283
Irreducibles, Primes......Page 285
Historical Discussion of Fermat\'s Last Theorem......Page 287
Unique Factorization Domains......Page 290
Euclidean Domains......Page 293
Exercises......Page 296
Biography of Ernst Eduard Kummer......Page 299
Biography of Sophie Germain......Page 300
Supplementary Exercises for Chapters 17-20......Page 301
Fields......Page 303
Definition and Examples......Page 305
Subspaces......Page 306
Linear Independence......Page 307
Exercises......Page 309
Biography of Emil Artin......Page 311
The Fundamental Theorem of Field Theory......Page 312
Splitting Fields......Page 314
Zeros of an Irreducible Polynomial......Page 319
Exercises......Page 321
Biography of Leopold Kronecker......Page 323
Characterization of Extensions......Page 324
Finite Extensions......Page 326
Properties of Algebraic Extensions......Page 330
Exercises......Page 331
Biography of Irving Kaplansky......Page 334
Classification of Finite Fields......Page 335
Structure of Finite Fields......Page 336
Subfields of a Finite Field......Page 339
Exercises......Page 341
Programming Exercises......Page 342
Biography ofL. E. Dickson......Page 343
Historical Discussion of Geometric Constructions......Page 344
Constructible Numbers......Page 345
Exercises......Page 347
Supplementary Exercises for Chapters 21-25......Page 350
Special Topics......Page 351
Conjugacy Classes......Page 353
The Class Equation......Page 354
The Probability That Two Elements Commute......Page 355
The Sylow Theorems......Page 356
Application of Sylow\'s Theorems......Page 359
Exercises......Page 364
Biography of Ludvig Sylow......Page 367
Historical Background......Page 368
Nonsimplicity Tests......Page 372
The Fields Medal......Page 376
Exercises......Page 377
Programming Exercises......Page 379
Biography of Michael Aschbacher......Page 381
Biography of Daniel Gorenstein......Page 382
Biography of John Thompson......Page 383
Motivation......Page 384
Definitions and Notation......Page 385
Free Group......Page 386
Generators and Relations......Page 387
Classification of Groups of Order up to 15......Page 390
Characterization of Dihedral Groups......Page 391
Realizing the Dihedral Groups with Mirrors......Page 392
Exercises......Page 394
Biography of William Burnside......Page 398
Isometries......Page 399
Classification of Finite Plane Symmetry Groups......Page 401
Classification of Finite Groups of Rotations in $\\mathbb{R}^3$......Page 402
Exercises......Page 405
The Frieze Groups......Page 408
The Crystallographic Groups......Page 413
Identification of Plane Periodic Patterns......Page 415
Exercises......Page 424
Biography of M. C. Escher......Page 429
Motivation......Page 430
The Cayley Digraph of a Group......Page 431
Hamiltonian Circuits and Paths......Page 434
Some Applications......Page 441
Exercises......Page 446
Biography of William Rowan Hamilton......Page 450
Biography of Paul Erdos......Page 452
Motivation......Page 454
Linear Codes......Page 459
Parity-Check Matrix Decoding......Page 463
Coset Decoding......Page 465
Exercises......Page 468
Biography of Richard W. Hamming......Page 473
Fundamental Theorem of Galois Theory......Page 474
Solvability of Polynomials by Radicals......Page 480
Insolvability of a Quintic......Page 483
Exercises......Page 484
Biography of Philip Hall......Page 487
Motivation......Page 488
Definition and Properties......Page 490
The Algebra of Electric Circuits......Page 492
The Algebra of Logic......Page 495
Finite Boolean Algebras......Page 496
Exercises......Page 497
Biography of Claude E. Shannon......Page 500
Supplementary Exercises for Chapters 26-34......Page 501
Selected Answers......Page 503
Notations......Page 537
Index of Mathematicians......Page 541
Index of Terms......Page 543
Back cover......Page 550