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