Abstract Algebra: A First Undergraduate Course

Description

Basic Set, Mapping, and Logic Notation Symbols Title Preface Contents Introduction 1. Basic Properties of the Integers 1.1. Mathematical Induction 1.2. Multiples and Divisors, Primes in Z 1.3. The Division Algorithm 1.4. Common Divisors 1.5. Euclid’s Algorithm 1.6. Common Multiples 1.7. Unique Factorization in Z 2. Groups 2.1. Permutations on a Finite Set 2.2. Multiplication of Permutations 2.3. Abstract Groups 2.4. Cycle Notation 2.5. Subgroups in a Group 2.6. Additive Notation, Modular Arithmetic 2.7. Cycllc Groups 2.8. Even and Odd Permutations 2.9. Groups of Symmetries 2.10. The Alternating Groups An 2.11. Cosets of a Subgroup 2.12. Quotient Groups 2.13. Solvable Groups 2.14. More on Symmetry 2.15. The Sylow Theorems (Without Proofs) 3. Sets and Mappings 3.1. Mappings 3.2. Group Isomorphisms 3.3. Group Homomorphisms 3.4. Cayley's Theorem 3.5. Unions, Intersections, Partitions 3.6. Cartesian Products, Direct Products 3.7. Relations 3.8. Partially Ordered Sets 3.9. Power Sets 3.10. Operations 3.11. Algebraic Structures 3.12. Boolean Algebra 3.13. Boolean Functions and Their Applications 3.14. Composition of Mappings, Groups of Bijections 3.15. Conjugacy Classes and the Class Equation 3.16. The Fundamental Theorem on Abelian Groups (Without Proof) 4. Rings and Fields 4.1. Rings 4.2. Ring Homomorphisms and Ideals 4.3. Congruence in Z, The Euler and Fermat Theorems 4.4. lntegral Domains 4.5. Fields 4.6. Ordered Integral Domains and Fields 4.7. Matrices, Quaternions 4.8. Embedding a Ring in a Field 4.9. Characterizations of Z, Q, R, and C 5. Polynomials 5.1. Polynomial Extensions of Rings 5.2. Polynomials over a Commutative Ring 5.3. Divisibility in Commutative Rings 5.4. Polynomial Functions 5.5. Integral and Rational Roots 5.6. Polynomial Fitting, Finite Differences 5.7. Ideals in F[x] 5.8. Factorizations of Polynomials 5.9. Partial Fractions 5.10. Extension Fields 5.11. Equations of Degree 2, 3, or 4 5.12. Automorphisms of E over F, lnsolvability of a Quintic 6. Euclidean Constructions 6.1. Closure Under Euclidean Constructions 6.2. Closure Under Square Roots 6.3. Constructible Points, Lines, and Circles 7. More on the Integers 7.1. Simultaneous Congruences -- The Chinese Remainder Theorem 7.2. More on Mathematical Induction 7.3. Some Number Theoretic Functions 7.4. Quadratic Residues 8. Some Applications to Coding 8.1. Binary Codes 8.2. Matrix Codes, A Hamming Code 8.3. Modular Codes, Trapdoor Functions Computer Programming Projects Supplementary and Challenging Problems Annotated Bibliography Answers, Hints, or Solutions for Most Odd-Numbered Problems Index Summary of Axioms for Groups, Rings, and Fields Some Group Tables