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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 |
Table of Contents provided by Ingram. All Rights Reserved. |