front cover -- copyright page -- title page -- Contents -- Preface -- Some Features of This Book -- A Note to Students -- A Note to Instructors -- Notation -- Early Number Theory -- Ancient Mathematics -- Diophantus -- Geometry and Pythagorean Triples -- The Method of Diophantus -- Fermat's Last Theorem -- Connections: Congruent Numbers -- Euclid -- Greek Number Theory -- Division and Remainders -- Linear Combinations and Euclid's Lemma -- Euclidean Algorithm -- Nine Fundamental Properties -- Connections -- Trigonometry -- Integration -- Induction -- Induction and Applications -- Unique Factorization -- Strong Induction -- Differential Equations -- Binomial Theorem -- Combinatorics -- Connections -- An Approach to Induction -- Fibonacci Sequence -- Renaissance -- Classical Formulas -- Complex Numbers -- Algebraic Operations -- Absolute Value and Direction -- The Geometry Behind Multiplication -- Roots and Powers -- Connections: Designing Good Problems -- Norms -- Pippins and Cheese -- Gaussian Integers: Pythagorean Triples Revisited -- Eisenstein Triples and Diophantus -- Nice Boxes -- Nice Functions for Calculus Problems -- Lattice Point Triangles -- Modular Arithmetic -- Congruence -- Public Key Codes -- Commutative Rings -- Units and Fields -- Subrings and Subfields -- Connections: Julius and Gregory -- Connections: Patterns in Decimal Expansions -- Real Numbers -- Decimal Expansions of Rationals -- Periods and Blocks -- Abstract Algebra -- Domains and Fraction Fields -- Polynomials -- Polynomial Functions -- Homomorphisms -- Extensions of Homomorphisms -- Kernel, Image, and Ideals -- Connections: Boolean Things -- Inclusion-Exclusion -- Arithmetic of Polynomials -- Parallels to Z -- Divisibility -- Roots -- Greatest Common Divisors -- Unique Factorization -- Principal Ideal Domains -- Irreducibility -- Roots of Unity.

Connections: Lagrange Interpolation -- Quotients, Fields, and Classical Problems -- Quotient Rings -- Field Theory -- Characteristics -- Extension Fields -- Algebraic Extensions -- Splitting Fields -- Classification of Finite Fields -- Connections: Ruler--Compass Constructions -- Constructing Regular n-gons -- Gauss's construction of the 17-gon -- Cyclotomic Integers -- Arithmetic in Gaussian and Eisenstein Integers -- Euclidean Domains -- Primes Upstairs and Primes Downstairs -- Laws of Decomposition -- Fermat's Last Theorem for Exponent 3 -- Preliminaries -- The First Case -- Gauss's Proof of the Second Case -- Approaches to the General Case -- Cyclotomic integers -- Kummer, Ideal Numbers, and Dedekind -- Connections: Counting Sums of Squares -- A Proof of Fermat's Theorem on Divisors -- Epilog -- Abel and Galois -- Solvability by Radicals -- Symmetry -- Groups -- Wiles and Fermat's Last Theorem -- Elliptic Integrals and Elliptic Functions -- Congruent Numbers Revisited -- Elliptic Curves -- Appendices -- Functions -- Equivalence Relations -- Vector Spaces -- Bases and Dimension -- Linear Transformations -- Inequalities -- Generalized Associativity -- A Cyclotomic Integer Calculator -- Eisenstein Integers -- Symmetric Polynomials -- Algebra with Periods -- References -- Index -- About the Authors.