A First Course in Abstract Algebra

Description

This text introduces readers to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.   Number Theory: Induction; Binomial Coefficients; Greatest Common Divisors; The Fundamental Theorem of Arithmetic Congruences; Dates and Days. Groups I: Some Set Theory; Permutations; Groups; Subgroups and Lagrange's Theorem; Homomorphisms; Quotient Groups; Group Actions; Counting with Groups. Commutative Rings I: First Properties; Fields; Polynomials; Homomorphisms; Greatest Common Divisors; Unique Factorization; Irreducibility; Quotient Rings and Finite Fields; Officers, Magic, Fertilizer, and Horizons. Linear Algebra: Vector Spaces; Euclidean Constructions; Linear Transformations; Determinants; Codes; Canonical Forms. Fields: Classical Formulas; Insolvability of the General Quintic; Epilog. Groups II: Finite Abelian Groups; The Sylow Theorems; Ornamental Symmetry. Commutative Rings III: Prime Ideals and Maximal Ideals; Unique Factorization; Noetherian Rings; Varieties; Grobner Bases.   For all readers interested in abstract algebra. Title Page Contents Preface 1 Number theory 1.1 Induction 1.2 Binomial Coefficients 1.3 Greatest Common Divisors 1.4 The Fundamental Theorem of Arithmetic 1.5 Congruences 1.6 Dates and Days 2 Groups I 2.1 Some Set Theory Functions Equivalence relations 2.2 Permutations 2.3 Groups Symmetry 2.4 Subgroups and Lagrange's Theorem 2.5 Homomorphisms 2.6 Quotient Groups 2.7 Group Actions 2.8 Counting with Groups 3 Rings I 3.1 First properties 3.2 Fields 3.3 Polynomials 3.4 Homomorphisms 3.5 Greatest Common Divisors Euclidean Rings 3.6 Unique Factorization 3.7 Irreducibility 3.8 Quotient Rings and Finite Fields 3.9 Officers, Magic, Fertilizer, and Horizons Officers Magic Fertilizer Horizons 4 Linear Algebra 4.1 Vector Spaces Gaussian Elimination 4.2 Euclidean Constructions 4.3 Linear Transformations 3.4 Determinants 3.5 Codes Block Codes Linear Codes 5 Fields 4.1 Classical Formulas Viète's Cubic Formula 4.2 Insolvability of the General Quintic Formulas and Solvability by Radicals Translation into Group Theory 5.3 Epilog 6 Groups II 6.1 Finite Abelian Groups 6.2 The Sylow Theorems 6.3 Ornamental Symmetry 7 Commutative Rings II 7.1 Prime Ideals and Maximal Ideals 7.2 Unique Factorization 7.3 Noetherian Rings 7.4 Varieties 7.5 Gröbner Bases Monomial Orders Generalized Division Algorithm Gröbner Bases Appendix A Appendix B Hints for Selected Exercises Bibliography Index