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Summary Table of Contents

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.

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Preface	p. ix
To the Instructor	p. xii
To the Student	p. xiv
Thematic Table of Contents for the Gore Course	p. xvi
The Core Course	p. 1
Arithmetic in Z Revisited	p. 3
The Division Algorithm	p. 3
Divisibility	p. 9
Primes and Unique Factorization	p. 17
Congruence in Z and Modular Arithmetic	p. 25
Congruence and Congruence Classes	p. 25
Modular Arithmetic	p. 32
The Structure of Z_p (p Prime) and Z_n	p. 37
Rings	p. 43
Definition and Examples of Rings	p. 44
Basic Properties of Rings	p. 59
Isomorphisms and Homomorphisms	p. 70
Arithmetic in F[x]	p. 85
Polynomial Arithmetic and the Division Algorithm	p. 86
Divisibility in F[x]	p. 95
Irreducibles and Unique Factorization	p. 100
Polynomial Functions, Roots, and Reducibility	p. 105
Irreducibility in Q[x]	p. 112
Irreducibility in R[x] and C[x]	p. 120
Congruence in F[x] and Congruence-Class Arithmetic	p. 125
Congruence in F[x] and Congruence Classes	p. 125
Congruence-Class Arithmetic	p. 130
The Structure of F[x]/(p(x)) When p(x) Is Irreducible	p. 135
Ideals and Quotient Rings	p. 141
Ideals and Congruence	p. 141
Quotient Rings and Homomorphisms	p. 152
The Structure of R/1 When / Is Prime or Maximal	p. 162
Groups	p. 169
Definition and Examples of Groups	p. 169
A Definition and Examples of Groups	p. 183
Basic Properties of Groups	p. 196
Subgroups	p. 203
Isomorphisms and Homomorphisms	p. 214
The Symmetric and Alternating Groups	p. 227
Normal Subgroups and Quotient Groups	p. 237
Congruence and Lagrange's Theorem	p. 237
Normal Subgroups	p. 248
Quotient Groups	p. 255
Quotient Groups and Homomorphisms	p. 263
The Simplicity of A_n	p. 273
Advanced Topics	p. 279
Topics in Group Theory	p. 281
Direct Products	p. 281
Finite Abelian Groups	p. 289
The Sylow Theorems	p. 298
Conjugacy and the Proof of the Sylow Theorems	p. 304
The Structure of Finite Groups	p. 312
Arithmetic in Integral Domains	p. 321
Euclidean Domains	p. 322
Principal Ideal Domains and Unique Factorization Domains	p. 332
Factorization of Quadratic Integers	p. 344
The Field of Quotients of an Integral Domain	p. 353
Unique Factorization in Polynomial Domains	p. 359
Field Extensions	p. 365
Vector Spaces	p. 365
Simple Extensions	p. 376
Algebraic Extensions	p. 382
Splitting Fields	p. 388
Separability	p. 394
Finite Fields	p. 399
Galois Theory	p. 407
The Galois Group	p. 407
The Fundamental Theorem of Galois Theory	p. 415
Solvability by Radicals	p. 423
Excursions and Applications	p. 435
Public-Key Cryptography	p. 437
Prerequisite: Section 2.3
The Chinese Remainder Theorem	p. 443
Proof of the Chinese Remainder Theorem	p. 443
Prerequisites: Section 2.1, Appendix C
Applications of the Chinese Remainder Theorem	p. 450
Prerequisite: Section 3.1
The Chinese Remainder Theorem for Rings	p. 453
Prerequisite: Section 6.2
Geometric Constructions	p. 459
Prerequisites: Sections 4.1, 4.4, and 4.5
Algebraic Coding Theory	p. 471
Linear Codes	p. 471
Prerequisites: Section 7.4, Appendix F
Decoding Techniques	p. 483
Prerequisite: Section 8.4
BCH Codes	p. 492
Prerequisite: Section 11.6
Appendices	p. 499
Logic and Proof	p. 500
Sets and Functions	p. 509
Well Ordering and Induction	p. 523
Equivalence Relations	p. 531
The Binomial Theorem	p. 537
Matrix Algebra	p. 540
Polynomials	p. 545
Bibliography	p. 553
Answers and Suggestions for Selected Odd-Numbered Exercises	p. 556
Index	p. 589
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