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
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