Introduction to Abstract Algebra

by Nicholson, W. Keith

• ISBN13: 9781118135358
• ISBN10: 1118135350
• Edition: 4th
• Format: Hardcover
• Copyright: 2012-03-20
• Publisher: Wiley

Summary

This Fourth Edition of  Introduction to Abstract Algebra  is a self-contained introduction to the basic structures of abstract algebra: groups, rings, and fields.   This book is intended for a one or two semester abstract algebra course.  The writing style is appealing to students, and great effort has been made to motivate and be very clear about how the topics and applications relate to one another.  Over 500 solved examples are included to aid reader comprehension as well as to demonstrate how results in the theory are obtained.  Many applications (particularly to coding theory, cryptography, and to combinatorics) are provided to illustrate how the abstract structures relate to real-world problems.  In addition, historical notes and biographies of mathematicians put the subject into perspective.  Abstract thinking is difficult when first encountered and this is addressed in this book by presenting concrete examples (induction, number theory, integers modulo n, permutations) before the abstract structures are defined.  With this approach, readers can complete computations immediately using concepts that will be seen again later in the abstract setting.  Special topics such as symmetric polynomials, nilpotent groups, and finite dimensional algebras are also discussed.

Author Biography

W. Keith Nicholson, PhD, is Professor in the Department of Mathematics and Statistics at the University of Calgary, Canada. He has published extensively in his areas of research interest, which include clean rings, morphic rings and modules, and quasi-morphic rings. Dr. Nicholson is the coauthor of Modern Algebra with Applications, Second Edition, also published by Wiley.

Table of Contents

Preface	p. ix
Acknowledgments	p. xvii
Notation Used in The Text	p. xix
A Sketch of the History of Algebra to 1929	p. xxiii
Preliminaries	p. 1
Proofs	p. 1
Sets	p. 5
Mappings	p. 9
Equivalences	p. 17
Integers and Permutations	p. 23
Induction	p. 24
Divisors and Prime Factorization	p. 32
Integers Modulo n	p. 42
Permutations	p. 53
An Application to Cryptography	p. 67
Groups	p. 69
Binary Operations	p. 70
Groups	p. 76
Subgroups	p. 86
Cyclic Groups and the Order of an Element	p. 90
Homomorphisms and Isomorphisms	p. 99
Cosets and Lagrange's Theorem	p. 108
Groups of Motions and Symmetries	p. 117
Normal Subgroups	p. 122
Factor Groups	p. 131
The Isomorphism Theorem	p. 137
An Application to Binary Linear Codes	p. 143
Rings	p. 159
Examples and Basic Properties	p. 160
Integral Domains and Fields	p. 171
Ideals and Factor Rings	p. 180
Homomorphisms	p. 189
Ordered Integral Domains	p. 199
Polynomials	p. 202
Polynomials	p. 203
Factorization of Polynomials Over a Field	p. 214
Factor Rings of Polynomials Over a Field	p. 227
Partial Fractions	p. 236
Symmetric Polynomials	p. 239
Formal Construction of Polynomials	p. 248
Factorization in Integral Domains	p. 251
Irreducibles and Unique Factorization	p. 252
Principal Ideal Domains	p. 264
Fields	p. 274
Vector Spaces	p. 275
Algebraic Extensions	p. 283
Splitting Fields	p. 291
Finite Fields	p. 298
Geometric Constructions	p. 304
The Fundamental Theorem of Algebra	p. 308
An Application to Cyclic and BCH Codes	p. 310
Modules over Principal Ideal Domains	p. 324
Modules	p. 324
Modules Over a PID	p. 335
p-Groups and the Sylow Theorems	p. 349
Products and Factors	p. 350
Cauchy's Theorem	p. 357
Group Actions	p. 364
The Sylow Theorems	p. 371
Semidirect Products	p. 379
An Application to Combinatorics	p. 382
Series of Subgroups	p. 388
The Jordan-Hölder Theorem	p. 389
Solvable Groups	p. 395
Nilpotent Groups	p. 401
Galois Theory	p. 412
Galois Groups and Separability	p. 413
The Main Theorem of Galois Theory	p. 422
Insolvability of Polynomials	p. 434
Cyclotomic Polynomials and Wedderburn's Theorem	p. 442
Finiteness Conditions for Rings and Modules	p. 447
Wedderburn's Theorem	p. 448
The Wedderburn-Artin Theorem	p. 457
Appendices	p. 471
Complex Numbers	p. 471
Matrix Algebra	p. 478
Zorn's Lemma	p. 486
Proof of the Recursion Theorem	p. 490
Bibliography	p. 492
Selected Answers	p. 495
Index	p. 523
Table of Contents provided by Ingram. All Rights Reserved.