Abstract Algebra An Introduction

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  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2012-07-27
  • Publisher: Cengage Learning

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ABSTRACT ALGEBRA: AN INTRODUCTION is intended for a first undergraduate course in modern abstract algebra. The flexible design of the text makes it suitable for courses of various lengths and different levels of mathematical sophistication, ranging from a traditional abstract algebra course to one with a more applied flavor. The emphasis is on clarity of exposition. The thematic development and organizational overview is what sets this book apart. The chapters are organized around three themes: arithmetic, congruence, and abstract structures. Each Them is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another.

Table of Contents

Prefacep. ix
To the Instructorp. xii
To the Studentp. xiv
Thematic Table of Contents for the Gore Coursep. xvi
The Core Coursep. 1
Arithmetic in Z Revisitedp. 3
The Division Algorithmp. 3
Divisibilityp. 9
Primes and Unique Factorizationp. 17
Congruence in Z and Modular Arithmeticp. 25
Congruence and Congruence Classesp. 25
Modular Arithmeticp. 32
The Structure of Zp (p Prime) and Znp. 37
Ringsp. 43
Definition and Examples of Ringsp. 44
Basic Properties of Ringsp. 59
Isomorphisms and Homomorphismsp. 70
Arithmetic in F[x]p. 85
Polynomial Arithmetic and the Division Algorithmp. 86
Divisibility in F[x]p. 95
Irreducibles and Unique Factorizationp. 100
Polynomial Functions, Roots, and Reducibilityp. 105
Irreducibility in Q[x]p. 112
Irreducibility in R[x] and C[x]p. 120
Congruence in F[x] and Congruence-Class Arithmeticp. 125
Congruence in F[x] and Congruence Classesp. 125
Congruence-Class Arithmeticp. 130
The Structure of F[x]/(p(x)) When p(x) Is Irreduciblep. 135
Ideals and Quotient Ringsp. 141
Ideals and Congruencep. 141
Quotient Rings and Homomorphismsp. 152
The Structure of R/1 When / Is Prime or Maximalp. 162
Groupsp. 169
Definition and Examples of Groupsp. 169
A Definition and Examples of Groupsp. 183
Basic Properties of Groupsp. 196
Subgroupsp. 203
Isomorphisms and Homomorphismsp. 214
The Symmetric and Alternating Groupsp. 227
Normal Subgroups and Quotient Groupsp. 237
Congruence and Lagrange's Theoremp. 237
Normal Subgroupsp. 248
Quotient Groupsp. 255
Quotient Groups and Homomorphismsp. 263
The Simplicity of Anp. 273
Advanced Topicsp. 279
Topics in Group Theoryp. 281
Direct Productsp. 281
Finite Abelian Groupsp. 289
The Sylow Theoremsp. 298
Conjugacy and the Proof of the Sylow Theoremsp. 304
The Structure of Finite Groupsp. 312
Arithmetic in Integral Domainsp. 321
Euclidean Domainsp. 322
Principal Ideal Domains and Unique Factorization Domainsp. 332
Factorization of Quadratic Integersp. 344
The Field of Quotients of an Integral Domainp. 353
Unique Factorization in Polynomial Domainsp. 359
Field Extensionsp. 365
Vector Spacesp. 365
Simple Extensionsp. 376
Algebraic Extensionsp. 382
Splitting Fieldsp. 388
Separabilityp. 394
Finite Fieldsp. 399
Galois Theoryp. 407
The Galois Groupp. 407
The Fundamental Theorem of Galois Theoryp. 415
Solvability by Radicalsp. 423
Excursions and Applicationsp. 435
Public-Key Cryptographyp. 437
Prerequisite: Section 2.3
The Chinese Remainder Theoremp. 443
Proof of the Chinese Remainder Theoremp. 443
Prerequisites: Section 2.1, Appendix C
Applications of the Chinese Remainder Theoremp. 450
Prerequisite: Section 3.1
The Chinese Remainder Theorem for Ringsp. 453
Prerequisite: Section 6.2
Geometric Constructionsp. 459
Prerequisites: Sections 4.1, 4.4, and 4.5
Algebraic Coding Theoryp. 471
Linear Codesp. 471
Prerequisites: Section 7.4, Appendix F
Decoding Techniquesp. 483
Prerequisite: Section 8.4
BCH Codesp. 492
Prerequisite: Section 11.6
Appendicesp. 499
Logic and Proofp. 500
Sets and Functionsp. 509
Well Ordering and Inductionp. 523
Equivalence Relationsp. 531
The Binomial Theoremp. 537
Matrix Algebrap. 540
Polynomialsp. 545
Bibliographyp. 553
Answers and Suggestions for Selected Odd-Numbered Exercisesp. 556
Indexp. 589
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