Introduction to Abstract Algebra / Edition 4

Introduction to Abstract Algebra / Edition 4

by W. Keith Nicholson
     
 

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ISBN-10: 1118135350

ISBN-13: 9781118135358

Pub. Date: 03/27/2012

Publisher: Wiley

Praise for the Third Edition

". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."—Zentralblatt MATH

The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra:

Overview

Praise for the Third Edition

". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."—Zentralblatt MATH

The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text.

The Fourth Edition features important concepts as well as specialized topics, including:

  • The treatment of nilpotent groups, including the Frattini and Fitting subgroups

  • Symmetric polynomials

  • The proof of the fundamental theorem of algebra using symmetric polynomials

  • The proof of Wedderburn's theorem on finite division rings

  • The proof of the Wedderburn-Artin theorem

Throughout the book, worked examples and real-world problems illustrate concepts and their applications, facilitating a complete understanding for readers regardless of their background in mathematics. A wealth of computational and theoretical exercises, ranging from basic to complex, allows readers to test their comprehension of the material. In addition, detailed historical notes and biographies of mathematicians provide context for and illuminate the discussion of key topics. A solutions manual is also available for readers who would like access to partial solutions to the book's exercises.

Introduction to Abstract Algebra, Fourth Edition is an excellent book for courses on the topic at the upper-undergraduate and beginning-graduate levels. The book also serves as a valuable reference and self-study tool for practitioners in the fields of engineering, computer science, and applied mathematics.

Product Details

ISBN-13:
9781118135358
Publisher:
Wiley
Publication date:
03/27/2012
Pages:
560
Sales rank:
638,786
Product dimensions:
7.20(w) x 10.10(h) x 1.30(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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