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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