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Preface | |
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To the Instructor | |
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To the Student | |
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Thematic Table of Contents for the Gore Course | |
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The Core Course | |
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Arithmetic in Z Revisited | |
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The Division Algorithm | |
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Divisibility | |
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Primes and Unique Factorization | |
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Congruence in Z and Modular Arithmetic | |
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Congruence and Congruence Classes | |
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Modular Arithmetic | |
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The Structure of Z<sub>p</sub> (p Prime) and Z<sub>n</sub> | |
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Rings | |
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Definition and Examples of Rings | |
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Basic Properties of Rings | |
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Isomorphisms and Homomorphisms | |
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Arithmetic in F[x] | |
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Polynomial Arithmetic and the Division Algorithm | |
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Divisibility in F[x] | |
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Irreducibles and Unique Factorization | |
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Polynomial Functions, Roots, and Reducibility | |
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Irreducibility in Q[x] | |
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Irreducibility in R[x] and C[x] | |
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Congruence in F[x] and Congruence-Class Arithmetic | |
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Congruence in F[x] and Congruence Classes | |
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Congruence-Class Arithmetic | |
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The Structure of F[x]/(p(x)) When p(x) Is Irreducible | |
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Ideals and Quotient Rings | |
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Ideals and Congruence | |
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Quotient Rings and Homomorphisms | |
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The Structure of R/1 When / Is Prime or Maximal | |
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Groups | |
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Definition and Examples of Groups | |
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A Definition and Examples of Groups | |
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Basic Properties of Groups | |
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Subgroups | |
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Isomorphisms and Homomorphisms | |
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The Symmetric and Alternating Groups | |
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Normal Subgroups and Quotient Groups | |
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Congruence and Lagrange's Theorem | |
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Normal Subgroups | |
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Quotient Groups | |
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Quotient Groups and Homomorphisms | |
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The Simplicity of A<sub>n</sub> | |
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Advanced Topics | |
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Topics in Group Theory | |
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Direct Products | |
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Finite Abelian Groups | |
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The Sylow Theorems | |
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Conjugacy and the Proof of the Sylow Theorems | |
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The Structure of Finite Groups | |
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Arithmetic in Integral Domains | |
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Euclidean Domains | |
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Principal Ideal Domains and Unique Factorization Domains | |
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Factorization of Quadratic Integers | |
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The Field of Quotients of an Integral Domain | |
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Unique Factorization in Polynomial Domains | |
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Field Extensions | |
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Vector Spaces | |
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Simple Extensions | |
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Algebraic Extensions | |
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Splitting Fields | |
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Separability | |
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Finite Fields | |
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Galois Theory | |
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The Galois Group | |
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The Fundamental Theorem of Galois Theory | |
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Solvability by Radicals | |
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Excursions and Applications | |
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Public-Key Cryptography | |
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Prerequisite: Section 2.3 | |
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The Chinese Remainder Theorem | |
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Proof of the Chinese Remainder Theorem | |
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Prerequisites: Section 2.1, Appendix C | |
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Applications of the Chinese Remainder Theorem | |
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Prerequisite: Section 3.1 | |
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The Chinese Remainder Theorem for Rings | |
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Prerequisite: Section 6.2 | |
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Geometric Constructions | |
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Prerequisites: Sections 4.1, 4.4, and 4.5 | |
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Algebraic Coding Theory | |
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Linear Codes | |
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Prerequisites: Section 7.4, Appendix F | |
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Decoding Techniques | |
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Prerequisite: Section 8.4 | |
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BCH Codes | |
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Prerequisite: Section 11.6 | |
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Appendices | |
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Logic and Proof | |
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Sets and Functions | |
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Well Ordering and Induction | |
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Equivalence Relations | |
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The Binomial Theorem | |
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Matrix Algebra | |
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Polynomials | |
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Bibliography | |
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Answers and Suggestions for Selected Odd-Numbered Exercises | |
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Index | |