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Introduction to Abstract Algebra / Edition 2
     

Introduction to Abstract Algebra / Edition 2

by W. Keith Nicholson
 

ISBN-10: 0471331090

ISBN-13: 9780471331094

Pub. Date: 07/28/1999

Publisher: Wiley

Clearly written and self-contained, this new edition of a successful book features precise, careful explanations of advanced algebra concepts such as groups, semi-groups, rings, modules, fields, and similar structures. It then goes a step farther and presents interesting motivational applications of algebraic ideas to cryptography, coding theory, etc.

Overview

Clearly written and self-contained, this new edition of a successful book features precise, careful explanations of advanced algebra concepts such as groups, semi-groups, rings, modules, fields, and similar structures. It then goes a step farther and presents interesting motivational applications of algebraic ideas to cryptography, coding theory, etc.

Product Details

ISBN-13:
9780471331094
Publisher:
Wiley
Publication date:
07/28/1999
Edition description:
Older Edition
Pages:
624
Product dimensions:
6.40(w) x 9.39(h) x 1.32(d)

Related Subjects

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

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