Abstract Algebra : an introduction

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Title:

Personal Author:

Hungerford, Thomas W.

Edition:

Third edition.

Publication Information:

Boston, MA : Brooks/Cole, Cengage Learning, 2014

Physical Description:

xvii, 595p. : illustrations ; 24 cm

ISBN:

9781111569624

General Note:

Includes index.

Subject Term:

Algebra, Abstract -- Textbooks.

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... MJIIT Library	30000010242858	QA162 H86 2012	Open Access Book	Book	Searching... Unknown

Abstract Algebra: An Introduction is set apart by its thematic development and organization. The chapters are organized around two themes: arithmetic and congruence. Each theme 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. New to this edition is a "groups first" option that enables those who prefer to cover groups before rings to do so easily.

Preface	p. ix
To the Instructor	p. xii
To the Student	p. xiv
Thematic Table of Contents for the Gore Course	p. xvi
Part 1 The Core Course	p. 1
Chapter 1 Arithmetic in Z Revisited	p. 3
1.1 The Division Algorithm	p. 3
1.2 Divisibility	p. 9
1.3 Primes and Unique Factorization	p. 17
Chapter 2 Congruence in Z and Modular Arithmetic	p. 25
2.1 Congruence and Congruence Classes	p. 25
2.2 Modular Arithmetic	p. 32
2.3 The Structure of Z p (p Prime) and Z n	p. 37
Chapter 3 Rings	p. 43
3.1 Definition and Examples of Rings	p. 44
3.2 Basic Properties of Rings	p. 59
3.3 Isomorphisms and Homomorphisms	p. 70
Chapter 4 Arithmetic in F[x]	p. 85
4.1 Polynomial Arithmetic and the Division Algorithm	p. 86
4.2 Divisibility in F[x]	p. 95
4.3 Irreducibles and Unique Factorization	p. 100
4.4 Polynomial Functions, Roots, and Reducibility	p. 105
4.5* Irreducibility in Q[x]	p. 112
4.6* Irreducibility in R[x] and C[x]	p. 120
Chapter 5 Congruence in F[x] and Congruence-Class Arithmetic	p. 125
5.1 Congruence in F[x] and Congruence Classes	p. 125
5.2 Congruence-Class Arithmetic	p. 130
5.3 The Structure of F[x]/(p(x)) When p(x) Is Irreducible	p. 135
Chapter 6 Ideals and Quotient Rings	p. 141
6.1 Ideals and Congruence	p. 141
6.2 Quotient Rings and Homomorphisms	p. 152
6.3 The Structure of R/1 When / Is Prime or Maximal	p. 162
Chapter 7 Groups	p. 169
7.1 Definition and Examples of Groups	p. 169
7.1 A Definition and Examples of Groups	p. 183
7.2 Basic Properties of Groups	p. 196
7.3 Subgroups	p. 203
7.4 Isomorphisms and Homomorphisms	p. 214
7.5* The Symmetric and Alternating Groups	p. 227
Chapter 8 Normal Subgroups and Quotient Groups	p. 237
8.1 Congruence and Lagrange's Theorem	p. 237
8.2 Normal Subgroups	p. 248
8.3 Quotient Groups	p. 255
8.4 Quotient Groups and Homomorphisms	p. 263
8.5 The Simplicity of A n	p. 273
Part 2 Advanced Topics	p. 279
Chapter 9 Topics in Group Theory	p. 281
9.1 Direct Products	p. 281
9.2 Finite Abelian Groups	p. 289
9.3 The Sylow Theorems	p. 298
9.4 Conjugacy and the Proof of the Sylow Theorems	p. 304
9.5 The Structure of Finite Groups	p. 312
Chapter 10 Arithmetic in Integral Domains	p. 321
10.1 Euclidean Domains	p. 322
10.2 Principal Ideal Domains and Unique Factorization Domains	p. 332
10.3 Factorization of Quadratic Integers	p. 344
10.4 The Field of Quotients of an Integral Domain	p. 353
10.5 Unique Factorization in Polynomial Domains	p. 359
Chapter 11 Field Extensions	p. 365
11.1 Vector Spaces	p. 365
11.2 Simple Extensions	p. 376
11.3 Algebraic Extensions	p. 382
11.4 Splitting Fields	p. 388
11.5 Separability	p. 394
11.6 Finite Fields	p. 399
Chapter 12 Galois Theory	p. 407
12.1 The Galois Group	p. 407
12.2 The Fundamental Theorem of Galois Theory	p. 415
12.3 Solvability by Radicals	p. 423
Part 3 Excursions and Applications	p. 435
Chapter 13 Public-Key Cryptography	p. 437
Prerequisite: Section 2.3
Chapter 14 The Chinese Remainder Theorem	p. 443
14.1 Proof of the Chinese Remainder Theorem	p. 443
Prerequisites: Section 2.1, Appendix C
14.2 Applications of the Chinese Remainder Theorem	p. 450
Prerequisite: Section 3.1
14.3 The Chinese Remainder Theorem for Rings	p. 453
Prerequisite: Section 6.2
Chapter 15 Geometric Constructions	p. 459
Prerequisites: Sections 4.1, 4.4, and 4.5
Chapter 16 Algebraic Coding Theory	p. 471
16.1 Linear Codes	p. 471
Prerequisites: Section 7.4, Appendix F
16.2 Decoding Techniques	p. 483
Prerequisite: Section 8.4
16.3 BCH Codes	p. 492
Prerequisite: Section 11.6
Part 4 Appendices	p. 499
A Logic and Proof	p. 500
B Sets and Functions	p. 509
C Well Ordering and Induction	p. 523
D Equivalence Relations	p. 531
E The Binomial Theorem	p. 537
F Matrix Algebra	p. 540
G Polynomials	p. 545
Bibliography	p. 553
Answers and Suggestions for Selected Odd-Numbered Exercises	p. 556
Index	p. 589

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