Special Notation Contents Preface to the Third Edition Chapter 1 Number Theory Section 1.1 Induction Section 1.2 Binomial Coefficients Section 1.3 Greatest Common Divisors Section 1.4 The Fundamental Theorem of Arithmetic Section 1.5 Congruences Section 1.6 Dates and Days Chapter 2 Groups I Section 2.1 Some Set Theory Section 2.2 Permutations Section 2.3 Groups Section 2.4 Subgroups and Lagrange’s Theorem Section 2.5 Homomorphisms Section 2.6 Quotient Groups . Section 2.7 Group Actions Section 2.8 Counting with Groups Chapter 3 Commutative Rings I Section 3.1 First Properties Section 3.2 Fields Section 3.3 Polynomials Section 3.4 Homomorphisms Section 3.5 Greatest Common Divisors Section 3.6 Unique Factorization Section 3.7 Irreducibility Section 3.8 Quotient Rings and Finite Fields Section 3.9 Officers, Magic, Fertilizer, and Horizons Chapter 4 Linear Algebra Section 4.1 Vector Spaces Section 4.2 Euclidean Constructions Section 4.3 Linear Transformations Section 4.4 Determinants Section 4.5 Codes Chapter 5 Fields Section 5.1 Classical Formulas Section 5.2 Insolvability of the General Quintic Section 5.3 Epilog Chapter 6 Groups II Section 6.1 Finite Abelian Groups Section 6.2 The Sylow Theorems Section 6.3 Ornamental Symmetry Chapter 7 Commutative Rings II Section 7.1 Prime Ideals and Maximal Ideals Section 7.2 Unique Factorization Section 7.3 Noetherian Rings Section 7.4 Varieties Section 7.5 Gr¨obner Bases Appendix A Inequalities Appendix B Pseudocodes Hints for Selected Exercises Bibliography Index