Introduction to Abstract Algebra, 3rd Edition

by W. Keith Nicholson (Univ. of Calgary, Canada)

• ISBN13: 9780471694922
• ISBN10: 0471694924
• Edition: 3rd
• Format: Hardcover
• Copyright: 2006-11-01
• Publisher: Wiley-Interscience

Summary

The Third Edition of Introduction to Abstract Algebra continues to provide an accessible introduction to the basic structures of abstract algebra: groups, rings, and fields. The text's unique approach helps you advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined.

Author Biography

W. KEITH NICHOLSON, PhD, is a Professor in the Department of Mathematics and Statistics at the University of Calgary in Alberta, Canada. Dr. Nicholson has published numerous textbooks and monographs throughout his career. He received his PhD in Pure Mathematics from the University of California at Santa Barbara, and his current research interests include clean rings, morphic rings, and module theory.

Table of Contents

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