Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- Part one The Kronecker - Duval Philosophy -- 1 Euclid -- 1.1 The Division Algorithm -- 1.2 Euclidean Algorithm -- 1.3 Bezout's Identity and Extended Euclidean Algorithm -- 1.4 Roots of Polynomials -- 1.5 Factorization of Polynomials -- 1.6 Computing a gcd -- 1.6.1 Coefficient explosion -- 1.6.2 Modular Algorithm -- 1.6.3 Hensel Lifting Algorithm -- 1.6.4 Heuristic gcd -- 2 Intermezzo: Chinese Remainder Theorems -- 2.1 Chinese Remainder Theorems -- 2.2 Chinese Remainder Theorem for a Principal Ideal Domain -- 2.3 A Structure Theorem (1) -- 2.4 Nilpotents -- 2.5 Idempotents -- 2.6 A Structure Theorem (2) -- 2.7 Lagrange Formula -- 3 Cardano -- 3.1 A Tautology? -- 3.2 The Imaginary Number -- 3.3 An Impasse -- 3.4 A Tautology! -- 4 Intermezzo: Multiplicity of Roots -- 4.1 Characteristic of a Field -- 4.2 Finite Fields -- 4.3 Derivatives -- 4.4 Multiplicity -- 4.5 Separability -- 4.6 Perfect Fields -- 4.7 Squarefree Decomposition -- 5 Kronecker I: Kronecker's Philosophy -- 5.1 Quotients of Polynomial Rings -- 5.2 The Invention of the Roots -- 5.3 Transcendental and Algebraic Field Extensions -- 5.4 Finite Algebraic Extensions -- 5.5 Splitting Fields -- 6 Intermezzo: Sylvester -- 6.1 Gauss Lemma -- 6.2 Symmetric Functions -- 6.3 Newton's Theorem -- 6.4 The Method of Indeterminate Coefficients -- 6.5 Discriminant -- 6.6 Resultants -- 6.7 Resultants and Roots -- 7 Galois I: Finite Fields -- 7.1 Galois Fields -- 7.2 Roots of Polynomials over Finite Fields -- 7.3 Distinct Degree Factorization -- 7.4 Roots of Unity and Primitive Roots -- 7.5 Representation and Arithmetics of Finite Fields -- 7.6 Cyclotomic Polynomials -- 7.7 Cycles, Roots and Idempotents -- 7.8 Deterministic Polynomial-time Primality Test -- 8 Kronecker II: Kronecker's Model -- 8.1 Kronecker's Philosophy.

8.2 Explicitly Given Fields -- 8.3 Representation and Arithmetics -- 8.3.1 Representation -- 8.3.2 Vector space arithmetics -- 8.3.3 Canonical representation -- 8.3.4 Multiplication -- 8.3.5 Inverse and division -- 8.3.6 Polynomial factorization -- 8.3.7 Solving polynomial equations -- 8.3.8 Monic polynomials -- 8.4 Primitive Element Theorems -- 9 Steinitz -- 9.1 Algebraic Closure -- 9.2 Algebraic Dependence and Transcendency Degree -- 9.3 The Structure of Field Extensions -- 9.4 Universal Field -- 9.5 Lüroth's Theorem -- 10 Lagrange -- 10.1 Conjugates -- 10.2 Normal Extension Fields -- 10.3 Isomorphisms -- 10.4 Splitting Fields -- 10.5 Trace and Norm -- 10.6 Discriminant -- 10.7 Normal Bases -- 11 Duval -- 11.1 Explicit Representation of Rings -- 11.2 Ring Operations in a Non-unique Representation -- 11.3 Duval Representation -- 11.4 Duval's Model -- 12 Gauss -- 12.1 The Fundamental Theorem of Algebra -- 12.2 Cyclotomic Equations -- 13 Sturm -- 13.1 Real Closed Fields -- 13.2 Definitions -- 13.3 Sturm -- 13.4 Sturm Representation of Algebraic Reals -- 13.5 Hermite's Method -- 13.6 Thom Codification of Algebraic Reals (1) -- 13.7 Ben-Or, Kozen and Reif Algorithm -- 13.8 Thom Codification of Algebraic Reals (2) -- 14 Galois II -- 14.1 Galois Extension -- 14.2 Galois Correspondence -- 14.3 Solvability by Radicals -- 14.4 Abel-Ruffini Theorem -- 14.5 Constructions with Ruler and Compass -- Part two Factorization -- 15 Prelude -- 15.1 A Computation -- 15.2 An Exercise -- 16 Kronecker III: factorization -- 16.1 Von Schubert Factorization Algorithm over the Integers -- 16.2 Factorization of Multivariate Polynomials -- 16.3 Factorization over a Simple Algebraic Extension -- 17 Berlekamp -- 17.1 Berlekamp's Algorithm -- 17.2 The Cantor-Zassenhaus Algorithm -- 18 Zassenhaus -- 18.1 Hensel's Lemma -- 18.2 The Zassenhaus Algorithm.

18.3 Factorization Over a Simple Transcendental Extension -- 18.4 Cauchy Bounds -- 18.5 Factorization over the Rationals -- 18.6 Swinnerton-Dyer Polynomials -- 18.7 L Algorithm -- 19 Finale -- 19.1 Kronecker's Dream -- 19.2 Van der Waerden's Example -- Bibliography -- Index.