Abstract algebra : an interactive approach

Cover......Page 1
Title Page......Page 5
Copyright......Page 6
Contents......Page 7
List of Figures......Page 11
List of Tables......Page 13
Preface......Page 15
Acknowledgments......Page 17
About the Author......Page 19
Symbol Description......Page 21
Mathematica® vs. GAP......Page 25
1.1 Introduction to Groups......Page 33
1.2 Modular Arithmetic......Page 37
1.3 Prime Factorizations......Page 42
1.4 The Definition of a Group......Page 47
Problems for Chapter 1......Page 53
2.1 Generators of Groups......Page 59
2.2 Defining Finite Groups in Mathematica and GAP......Page 63
2.3 Subgroups......Page 70
Problems for Chapter 2......Page 80
3.1 Left and Right Cosets......Page 85
3.2 How to Write a Secret Message......Page 90
3.3 Normal Subgroups......Page 98
3.4 Quotient Groups......Page 103
Problems for Chapter 3......Page 106
4.1 Isomorphisms......Page 111
4.2 Homomorphisms......Page 118
4.3 The Three Isomorphism Theorems......Page 125
Problems for Chapter 4......Page 135
5.1 Symmetric Groups......Page 139
5.2 Cycles......Page 143
5.3 Cayley's Theorem......Page 153
5.4 Numbering the Permutations......Page 159
Problems for Chapter 5......Page 162
6.1 The Direct Product......Page 167
6.2 The Fundamental Theorem of Finite Abelian Groups......Page 173
6.3 Automorphisms......Page 183
6.4 Semi-Direct Products......Page 193
Problems for Chapter 6......Page 203
7.1 The Center of a Group......Page 207
7.2 The Normalizer and Normal Closure Subgroups......Page 211
7.3 Conjugacy Classes and Simple Groups......Page 215
7.4 The Class Equation and Sylow's Theorems......Page 222
Problems for Chapter 7......Page 235
8.1 Subnormal Series and the Jordan-Hölder Theorem......Page 241
8.2 Derived Group Series......Page 249
8.3 Polycyclic Groups......Page 256
8.4 Solving the Pyraminx[sup(TM)]......Page 264
Problems for Chapter 8......Page 271
9.1 Groups with an Additional Operation......Page 277
9.2 The Definition of a Ring......Page 284
9.3 Entering Finite Rings into GAP and Mathematica......Page 288
9.4 Some Properties of Rings......Page 296
Problems for Chapter 9......Page 301
10.1 Subrings......Page 305
10.2 Quotient Rings and Ideals......Page 309
10.3 Ring Isomorphisms......Page 316
10.4 Homomorphisms and Kernels......Page 324
Problems for Chapter 10......Page 334
11.1 Polynomial Rings......Page 341
11.2 The Field of Quotients......Page 350
11.3 Complex Numbers......Page 356
11.4 Ordered Commutative Rings......Page 370
Problems for Chapter 11......Page 377
12.1 Factorization of Polynomials......Page 383
12.2 Unique Factorization Domains......Page 394
12.3 Principal Ideal Domains......Page 405
12.4 Euclidean Domains......Page 411
Problems for Chapter 12......Page 417
13.1 Entering Finite Fields in Mathematica or GAP......Page 423
13.2 Properties of Finite Fields......Page 428
13.3 Cyclotomic Polynomials......Page 437
13.4 Finite Skew Fields......Page 449
Problems for Chapter 13......Page 455
14.1 Vector Spaces......Page 461
14.2 Extension Fields......Page 468
14.3 Splitting Fields......Page 476
Problems for Chapter 14......Page 487
15.1 The Galois Group of an Extension Field......Page 491
15.2 The Galois Group of a Polynomial in Q......Page 500
15.3 The Fundamental Theorem of Galois Theory......Page 511
15.4 Solutions of Polynomial Equations Using Radicals......Page 518
Problems for Chapter 15......Page 523
Answers to Odd-Numbered Problems......Page 529
Bibliography......Page 549