Abstract Algebra: An Introduction

Cover......Page 1
Notations......Page 2
Contents......Page 9
Preface......Page 13
To The Instructor......Page 16
To The Student......Page 18
Thematic Table of Contents for the Core Course......Page 20
Part 1 The Core Course......Page 23
1.1 The Division Algorithm......Page 25
1.2 Divisibility......Page 31
1.3 Primes and Unique Factorization......Page 39
2.1 Congruence and Congruence Classes......Page 47
2.2 Modular Arithmetic......Page 54
2.3 The Structure of ZP (p Prime) and Zn......Page 59
CHAPTER 3 Rings......Page 65
3.1 Definition and Examples of Rings......Page 66
3.2 Basic Properties of Rings......Page 81
3.3 Isomorphisms and Homomorphisms......Page 92
CHAPTER 4 Arithmetic in f[x]......Page 107
4.1 Polynomial Arithmetic and the Division Algorithm......Page 108
4.2 Divisibility in F[x]......Page 117
4.3 lrreducibles and Unique Factorization......Page 122
4.4 Polynomial Functions, Roots, and Reducibility......Page 127
4.5 Irreducibility in Q[x]*......Page 134
4.6 Irreducibility in R[x] and C[x]*......Page 142
5.1 Congruence in F[x] and Congruence Classes......Page 147
5.2 Congruence-Class Arithmetic......Page 152
5.3 The Structure of F[x]/(p(x)) When p(x) Is Irreducible......Page 157
6.1 Ideals and Congruence......Page 163
6.2 Quotient Rings and Homomorphisms......Page 174
6.3 The Structure of R//When /Is Prime or Maximal*......Page 184
7.1 Definition and Examples of Groups......Page 191
7.2 Basic Properties of Groups......Page 218
7.3 Subgroups......Page 225
7.4 Isomorphisms and Homomorphisms*......Page 236
7.5 The Symmetric and Alternating Groups*......Page 249
8.1 Congruence and Lagrange's Theorem......Page 259
8.2 Normal Subgroups......Page 270
8.3 Quotient Groups......Page 277
8.4 Quotient Groups and Homomorphisms......Page 285
II The Simplicity of An*......Page 295
Part 2 Advanced Topics......Page 301
9.1 Direct Products......Page 303
9.2 Finite Abelian Groups......Page 311
9.3 The Sylow Theorems......Page 320
9.4 Conjugacy and the Proof of the Sylow Theorems......Page 326
9.5 The Structure of Finite Groups......Page 334
CHAPTER 10 Arithmetic in Integral Domains......Page 343
10.1 Euclidean Domains......Page 344
10.2 Principal Ideal Domains and Unique FactorizationDomains......Page 354
10.3 Factorization of Quadratic Integers*......Page 366
10.4 The Field of Quotients of an Integral Domain*......Page 375
10.5 Unique Factorization in Polynomial Domains*......Page 381
11.1 Vector Spaces......Page 387
11.2 Simple Extensions......Page 398
11.3 Algebraic Extensions......Page 404
11.4 Splitting Fields......Page 410
11.5 Separability......Page 416
11.6 Finite Fields......Page 421
12.1 The Galois Group......Page 429
12.2 The Fundamental Theorem of Galois Theory......Page 437
12.3 Solvability by Radicals......Page 445
Part 3 Excursions and Applications......Page 457
CHAPTER 13 Public-Key Cryptography......Page 459
14.1 Proof of the Chinese Remainder Theorem......Page 465
14.2 Applications of the Chinese Remainder Theorem......Page 472
14.3 The Chinese Remainder Theorem for Rings......Page 475
CHAPTER 15 Geometric Constructions......Page 481
16.1 Linear Codes......Page 493
16.2 Decoding Techniques......Page 505
16.3 BCH Codes......Page 514
Part 4 Appendices......Page 521
A. Logic and Proof......Page 522
B. Sets and Functions......Page 531
C. Well Ordering and Induction......Page 545
D. Equivalence Relations......Page 553
E. The Binomial Theorem......Page 559
F. Matrix Algebra......Page 562
6. Polynomials......Page 567
Bibliography......Page 575
Answers and Suggestions for Selected Odd-Numbered......Page 578
Index......Page 611