ELEMENTS OF MODERN ALGEBRA 7e, with its user-friendly format, provides you with the tools you need to get succeed in abstract algebra and develop mathematical maturity as a bridge to higher-level mathematics courses.. Strategy boxes give you guidance and explanations about techniques and enable you to become more proficient at constructing proofs. A summary of key words and phrases at the end of each chapter help you master the material. A reference section, symbolic marginal notes, an appendix, and numerous examples help you develop your problem solving skills.
Author(s): Linda Gilbert, Jimmie Gilbert
Edition: 7th Edition
Publisher: Brooks Cole
Year: 2009
Language: English
Pages: 524
Tags: Математика;Общая алгебра;
Front Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 8
Preface......Page 12
1.1 Sets......Page 18
Exercises 1.1......Page 26
1.2 Mappings......Page 29
Exercises 1.2......Page 38
1.3 Properties of Composite Mappings (Optional)......Page 42
Exercises 1.3......Page 45
1.4 Binary Operations......Page 47
Exercises 1.4......Page 51
1.5 Permutations and Inverses......Page 54
Exercises 1.5......Page 58
1.6 Matrices......Page 59
Exercises 1.6......Page 68
1.7 Relations......Page 72
Exercises 1.7......Page 75
A Pioneer in Mathematics: Arthur Cayley......Page 79
2.1 Postulates for the Integers (Optional)......Page 82
Exercises 2.1......Page 86
2.2 Mathematical Induction......Page 88
Exercises 2.2......Page 93
2.3 Divisibility......Page 98
Exercises 2.3......Page 101
2.4 Prime Factors and Greatest Common Divisor......Page 103
Exercises 2.4......Page 109
2.5 Congruence of Integers......Page 112
Exercises 2.5......Page 120
2.6 Congruence Classes......Page 124
Exercises 2.6......Page 129
2.7 Introduction to Coding Theory (Optional)......Page 131
Exercises 2.7......Page 136
2.8 Introduction to Cryptography (Optional)......Page 140
Exercises 2.8......Page 147
Key Words and Phrases......Page 151
A Pioneer in Mathematics: Blaise Pascal......Page 152
3.1 Definition of a Group......Page 154
Exercises 3.1......Page 158
3.2 Properties of Group Elements......Page 162
Exercises 3.2......Page 167
3.3 Subgroups......Page 169
Exercises 3.3......Page 176
3.4 Cyclic Groups......Page 180
Exercises 3.4......Page 187
3.5 Isomorphisms......Page 191
Exercises 3.5......Page 197
3.6 Homomorphisms......Page 200
Exercises 3.6......Page 203
Key Words and Phrases......Page 205
A Pioneer in Mathematics: Niels Henrik Abel......Page 206
4.1 Finite Permutation Groups......Page 208
Exercises 4.1......Page 219
4.2 Cayley's Theorem......Page 222
Exercises 4.2......Page 224
4.3 Permutation Groups in Science and Art (Optional)......Page 225
Exercises 4.3......Page 229
4.4 Cosets of a Subgroup......Page 232
Exercises 4.4......Page 237
4.5 Normal Subgroups......Page 240
Exercises 4.5......Page 244
4.6 Quotient Groups......Page 247
Exercises 4.6......Page 253
4.7 Direct Sums (Optional)......Page 256
Exercises 4.7......Page 261
4.8 Some Results on Finite Abelian Groups (Optional)......Page 263
Exercises 4.8......Page 271
Key Words and Phrases......Page 272
A Pioneer in Mathematics: Augustin Louis Cauchy......Page 273
5.1 Definition of a Ring......Page 274
Exercises 5.1......Page 282
5.2 Integral Domains and Fields......Page 287
Exercises 5.2......Page 290
5.3 The Field of Quotients of an Integral Domain......Page 293
Exercises 5.3......Page 299
5.4 Ordered Integral Domains......Page 301
Exercises 5.4......Page 306
Key Words and Phrases......Page 308
A Pioneer in Mathematics: Richard Dedekind......Page 309
6.1 Ideals and Quotient Rings......Page 310
Exercises 6.1......Page 317
6.2 Ring Homomorphisms......Page 320
Exercises 6.2......Page 326
6.3 The Characteristic of a Ring......Page 330
Exercises 6.3......Page 334
6.4 Maximal Ideals (Optional)......Page 336
Exercises 6.4......Page 339
A Pioneer in Mathematics: Amalie Emmy Noether......Page 341
7.1 The Field of Real Numbers......Page 342
Exercises 7.1......Page 349
7.2 Complex Numbers and Quaternions......Page 350
Exercises 7.2......Page 357
7.3 De Moivre's Theorem and Roots of Complex Numbers......Page 360
Exercises 7.3......Page 366
Key Words and Phrases......Page 369
A Pioneer in Mathematics: William Rowan Hamilton......Page 370
8.1 Polynomials over a Ring......Page 372
Exercises 8.1......Page 381
8.2 Divisibility and Greatest Common Divisor......Page 384
Exercises 8.2......Page 390
8.3 Factorization in F[x]......Page 392
Exercises 8.3......Page 398
8.4 Zeros of a Polynomial......Page 401
Exercises 8.4......Page 411
8.5 Solution of Cubic and Quartic Equations by Formulas (Optional)......Page 414
Exercises 8.5......Page 425
8.6 Algebraic Extensions of a Field......Page 426
Exercises 8.6......Page 436
Key Words and Phrases......Page 438
A Pioneer in Mathematics: Carl Friedrich Gauss......Page 439
APPENDIX: The Basics of Logic......Page 440
Answers to True/False and Selected Computational Exercises......Page 452
Bibliography......Page 516
Index......Page 520
