1 Euclid's Algorithm -- 1.1 Euclidean Algorithm -- 1.2 Diophantine Approximations -- 1.3 Continued Fractions -- 1.4 Diophantine Equations -- 2 Continued Fractions -- 2.1 Basics -- 2.2 Matrix Representation -- 2.3 Continuant Representation -- 2.4 Continued Fractions of Quadratics -- 2.5 Approximation Properties -- 2.6 Continued Fraction Arithmetic -- 3 Diophantine Equations -- 3.1 Two Variable Linear Diophantine Equations -- 3.2 General Linear Diophantine Equations -- 3.3 Pell's Equation -- 3.4 Fermat's Last Theorem -- 4 Lattice Techniques -- 4.1 Lattice Fundamentals -- 4.2 Minkowski Convex Body Theorem -- 4.3 Reduced Bases -- 4.4 Finding Numerical Relationships -- 5 Arithmetic Functions -- 5.1 Arithmetic Functions -- 5.2 Asymptotic Behavior of Arithmetic Functions -- 5.3 Distribution of Primes -- 5.4 Bertrand's Postulate -- 6 Residue Rings -- 6.1 Basic Properties of ?/m? -- 6.2 Chinese Remainder Theorem -- 6.3 Multiplicative Structure of ?/m? -- 6.4 Quadratic Reciprocity -- 6.5 Algebraic Extensions of -- 6.6 p-adic Numbers -- 6.7 Cryptosystems -- 6.8 Sums of Squares -- 7 Polynomial Arithmetic -- 7.1 Generalities -- 7.2 Polynomial Addition -- 7.3 Polynomial Multiplication -- 7.4 Fast Polynomial Algorithms -- 7.5 Polynomial Exponentiation -- 7.6 Polynomial Substitution -- 8 Polynomial GCD's: Classical Algorithms -- 8.1 Generalities -- 8.2 GCD of Several Quantities -- 8.3 Polynomial Contents -- 8.4 Coefficient Growth -- 8.5 Pseudo-Quotients -- 8.6 Subresultant Polynomial Remainder Sequence -- 9 Polynomial Elimination -- 9.1 Symmetric Functions -- 9.2 Polynomial Resultants -- 9.3 Subresultants -- 9.4 Elimination Examples -- 10 Formal Power Series -- 10.1 Introduction -- 10.2 Power Series Arithmetic -- 10.3 Power Series Exponentiation -- 10.4 Composition of Formal Power Series -- 10.5 Reversion of Power Series -- 11 Bounds on Polynomials -- 11.1 Heights of Polynomials -- 11.2 Uniform Coefficient Bounds -- 11.3 Weighted Coefficient Bounds -- 11.4 Size of a Polynomial's Zeroes -- 11.5 Discriminants and Zero Separation -- 12 Zero Equivalence Testing -- 12.1 Probabilistic Techniques -- 12.2 Deterministic Results -- 12.3 Negative Results -- 13 Univariate Interpolation -- 13.1 Vandermonde Matrices -- 13.2 Lagrange Interpolation -- 13.3 Newton Interpolation -- 13.4 Fast Fourier Transform -- 13.5 Abstract Interpolation -- 14 Multivariate Interpolation -- 14.1 Multivariate Dénse Interpolation -- 14.2 Probabilistic Sparse Interpolation -- 14.3 Deterministic Sparse Interpolation with Degree Bounds -- 14.4 Deterministic Sparse Interpolation without Degree Bounds -- 15 Polynomial GCD's: Interpolation Algorithms -- 15.1 Heuristic GCD -- 15.2 Univariate Polynomials over ? -- 15.3 Multivariate Polynomials -- 16 Hensel Algorithms -- 16.1 m-adic Completions -- 16.2 One Dimensional Iteration -- 16.3 Multidimensional Iteration -- 16.4 Hensel's Lemma -- 16.5 Generalizations of Hensel's Lemma -- 16.6 Zassenhaus' Formulation of Hensel's Lemma -- 17 Sparse Hensel Algorithms -- 17.1 Heuristic Presentation -- 17.2 Formal Presentation -- 18 Factoring over Finite Fields -- 18.1 Square Free Decomposition -- 18.2 Distinct Degree Factorization -- 18.3 Finding Linear Factors -- 18.4 Cantor-Zassenhaus Algorithm -- 19 Irreducibility of Polynomials -- 19.1 Deterministic Irreducibility Testing -- 19.2 Counting Prime Factors -- 19.3 Hilbert Irreducibility Theorem -- 19.4 Bertini's Theorem -- 20 Univariate Factorization -- 20.1 Reductions -- 20.2 Simple Algorithm -- 20.3 Asymptotically Good Algorithms -- 21 Multivariate Factorization -- 21.1 General Reductions -- 21.2 Lifting Multivariate Factorizations -- 21.3 Leading Coefficient Determination -- 21.4 Multivariate Polynomials over Q -- 21.5 Bivariate Polynomials over Fields -- List of symbols.