The p
Trade Review
“This could also be an excellent adjunct to more theoretically oriented textbooks used in more intensive courses.” (Computing Reviews, 5 November 2012)
Table of Contents
Preface ix Acknowledgment xv
Notations Used in the Text xvii
A Sketch of the History of Algebra to 1929 xxi
Preliminaries 1
Proofs 1
Sets 5
Mappings 9
Equivalences 17
Integers and Permutations 22
Induction 22
Divisors and Prime Factorization 30
Integers Modulo n 41
Permutations 51
An Application to Cryptography 63
Groups 66
Binary Operations 66
Groups 73
Subgroups 82
Cyclic Groups and the Order of an Element 87
Homomorphisms and Isomorphisms 95
Cosets and Lagrange's Theorem 105
Groups of Motions and Symmetries 114
Normal Subgroups 119
Factor Groups 127
The Isomorphism Theorem 133
An Application to Binary Linear Codes 140
Rings 155
Examples and Basic Properties 155
Integral Domains and Fields 166
Ideals and Factor Rings 174
Homomorphisms 183
Ordered Integral Domains 193
Polynomials 196
Polynomials 196
Factorization of Polynomials over a Field 209
Factor Rings of Polynomials over a Field 222
Partial Fractions 231
Symmetric Polynomials 233
Formal Construction of Polynomials 243
Factorization in Integral Domains 246
Irreducibles and Unique Factorization 247
Principal Ideal Domains 259
Fields 268
Vector Spaces 269
Algebraic Extensions 277
Splitting Fields 285
Finite Fields 293
Geometric Constructions 299
The Fundamental Theorem of Algebra 304
An Application to Cyclic and BCH Codes 305
Modules over Principal Ideal Domains 318
Modules 318
Modules over a PID 327
p-Groups and the Sylow Theorems 341
Factors and Products 341
Cauchy's Theorem 349
Group Actions 356
The Sylow Theorems 364
Semidirect Products 371
An Application to Combinatorics 375
Series of Subgroups 381
The Jordan-Holder Theorem 382
Solvable Groups 387
Nilpotent Groups 394
Galois Theory 401
Galois Groups and Separability 402
The Main Theorem of Galois Theory 410
Insolvability of Polynomials 423
Cyclotomic Polynomials and Wedderburn's Theorem 430
Finiteness Conditions for Rings and Modules 435
Wedderburn's Theorem 435
The Wedderburn-Artin Theorem 444
Appendices
Complex Numbers 455
Matrix Arithmetic 462
Zorn's Lemma 467
Proof of the Recursion Theorem 471
Bibliography 473
Selected Answers 475
Index 499
