Introduction to Abstract Algebra

PREFACE ix

ACKNOWLEDGMENTS xvii

NOTATION USED IN THE TEXT xix

A SKETCH OF THE HISTORY OF ALGEBRA TO 1929 xxiii

0 Preliminaries 1

0.1 Proofs / 1

0.2 Sets / 5

0.3 Mappings / 9

0.4 Equivalences / 17

1 Integers and Permutations 23

1.1 Induction / 24

1.2 Divisors and Prime Factorization / 32

1.3 Integers Modulo n / 42

1.4 Permutations / 53

1.5 An Application to Cryptography / 67

2 Groups 69

2.1 Binary Operations / 70

2.2 Groups / 76

2.3 Subgroups / 86

2.4 Cyclic Groups and the Order of an Element / 90

2.5 Homomorphisms and Isomorphisms / 99

2.6 Cosets and Lagrange?s Theorem / 108

2.7 Groups of Motions and Symmetries / 117

2.8 Normal Subgroups / 122

2.9 Factor Groups / 131

2.10 The Isomorphism Theorem / 137

2.11 An Application to Binary Linear Codes / 143

3 Rings 159

3.1 Examples and Basic Properties / 160

3.2 Integral Domains and Fields / 171

3.3 Ideals and Factor Rings / 180

3.4 Homomorphisms / 189

3.5 Ordered Integral Domains / 199

4 Polynomials 202

4.1 Polynomials / 203

4.2 Factorization of Polynomials Over a Field / 214

4.3 Factor Rings of Polynomials Over a Field / 227

4.4 Partial Fractions / 236

4.5 Symmetric Polynomials / 239

4.6 Formal Construction of Polynomials / 248

5 Factorization in Integral Domains 251

5.1 Irreducibles and Unique Factorization / 252

5.2 Principal Ideal Domains / 264

6 Fields 274

6.1 Vector Spaces / 275

6.2 Algebraic Extensions / 283

6.3 Splitting Fields / 291

6.4 Finite Fields / 298

6.5 Geometric Constructions / 304

6.6 The Fundamental Theorem of Algebra / 308

6.7 An Application to Cyclic and BCH Codes / 310

7 Modules over Principal Ideal Domains 324

7.1 Modules / 324

7.2 Modules Over a PID / 335

8 p-Groups and the Sylow Theorems 349

8.1 Products and Factors / 350

8.2 Cauchy?s Theorem / 357

8.3 Group Actions / 364

8.4 The Sylow Theorems / 371

8.5 Semidirect Products / 379

8.6 An Application to Combinatorics / 382

9 Series of Subgroups 388

9.1 The Jordan?H¨older Theorem / 389

9.2 Solvable Groups / 395

9.3 Nilpotent Groups / 401

10 Galois Theory 412

10.1 Galois Groups and Separability / 413

10.2 The Main Theorem of Galois Theory / 422

10.3 Insolvability of Polynomials / 434

10.4 Cyclotomic Polynomials and Wedderburn?s Theorem / 442

11 Finiteness Conditions for Rings and Modules 447

11.1 Wedderburn?s Theorem / 448

11.2 The Wedderburn?Artin Theorem / 457

Appendices 471

Appendix A Complex Numbers / 471

Appendix B Matrix Algebra / 478

Appendix C Zorn?s Lemma / 486

Appendix D Proof of the Recursion Theorem / 490

BIBLIOGRAPHY 492

SELECTED ANSWERS 495

INDEX 523