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1. Fundamental Results and Algorithms in Dedekind Domains.- 1.1 Introduction.- 1.2 Finitely Generated Modules Over Dedekind Domains.- 1.2.1 Finitely Generated Torsion-Free and Projective Modules.- 1.2.2 Torsion Modules.- 1.3 Basic Algorithms in Dedekind Domains.- 1.3.1 Extended Euclidean Algorithms in Dedekind Domains.- 1.3.2 Deterministic Algorithms for the Approximation Theorem.- 1.3.3 Probabilistic Algorithms.- 1.4 The Hermite Normal Form Algorithm in Dedekind Domains.- 1.4.1 Pseudo-Objects.- 1.4.2 The Hermite Normal Form in Dedekind Domains.- 1.4.3 Reduction Modulo an Ideal.- 1.5 Applications of the HNF Algorithm.- 1.5.1 Modifications to the HNF Pseudo-Basis.- 1.5.2 Operations on Modules and Maps.- 1.5.3 Reduction Modulo p of a Pseudo-Basis.- 1.6 The Modular HNF Algorithm in Dedekind Domains.- 1.6.1 Introduction.- 1.6.2 The Modular HNF Algorithm.- 1.6.3 Computing the Transformation Matrix.- 1.7 The Smith Normal Form Algorithm in Dedekind Domains.- 1.8 Exercises for Chapter 1.- 2. Basic Relative Number Field Algorithms.- 2.1 Compositum of Number Fields and Relative and Absolute Equations.- 2.1.1 Introduction.- 2.1.2 Etale Algebras.- 2.1.3 Compositum of Two Number Fields.- 2.1.4 Computing (?1 and ?2.- 2.1.5 Relative and Absolute Defining Polynomials.- 2.1.6 Compositum with Normal Extensions.- 2.2 Arithmetic of Relative Extensions.- 2.2.1 Relative Signatures.- 2.2.2 Relative Norm, Trace, and Characteristic Polynomial.- 2.2.3 Integral Pseudo-Bases.- 2.2.4 Discriminants.- 2.2.5 Norms of Ideals in Relative Extensions.- 2.3 Representation and Operations on Ideals.- 2.3.1 Representation of Ideals.- 2.3.2 Representation of Prime Ideals.- 2.3.3 Computing Valuations.- 2.3.4 Operations on Ideals.- 2.3.5 Ideal Factorization and Ideal Lists.- 2.4 The Relative Round 2 Algorithm and Related Algorithms.- 2.4.1 The Relative Round 2 Algorithm.- 2.4.2 Relative Polynomial Reduction.- 2.4.3 Prime Ideal Decomposition.- 2.5 Relative and Absolute Representations.- 2.5.1 Relative and Absolute Discriminants.- 2.5.2 Relative and Absolute Bases.- 2.5.3 Ups and Downs for Ideals.- 2.6 Relative Quadratic Extensions and Quadratic Forms.- 2.6.1 Integral Pseudo-Basis, Discriminant.- 2.6.2 Representation of Ideals.- 2.6.3 Representation of Prime Ideals.- 2.6.4 Composition of Pseudo-Quadratic Forms.- 2.6.5 Reduction of Pseudo-Quadratic Forms.- 2.7 Exercises for Chapter 2.- 3. The Fundamental Theorems of Global Class Field Theory.- 3.1 Prologue: Hilbert Class Fields.- 3.2 Ray Class Groups.- 3.2.1 Basic Definitions and Notation.- 3.3 Congruence Subgroups: One Side of Class Field Theory.- 3.3.1 Motivation for the Equivalence Relation.- 3.3.2 Study of the Equivalence Relation.- 3.3.3 Characters of Congruence Subgroups.- 3.3.4 Conditions on the Conductor and Examples.- 3.4 Abelian Extensions: The Other Side of Class Field Theory.- 3.4.1 The Conductor of an Abelian Extension.- 3.4.2 The Frobenius Homomorphism.- 3.4.3 The Artin Map and the Artin Group Am(L/K).- 3.4.4 The Norm Group (or Takagi Group) Tm(L/K).- 3.5 Putting Both Sides Together: The Takagi Existence Theorem 154.- 3.5.1 The Takagi Existence Theorem.- 3.5.2 Signatures, Characters, and Discriminants.- 3.6 Exercises for Chapter 3.- 4. Computational Class Field Theory.- 4.1 Algorithms on Finite Abelian groups.- 4.1.1 Algorithmic Representation of Groups.- 4.1.2 Algorithmic Representation of Subgroups.- 4.1.3 Computing Quotients.- 4.1.4 Computing Group Extensions.- 4.1.5 Right Four-Term Exact Sequences.- 4.1.6 Computing Images, Inverse Images, and Kernels.- 4.1.7 Left Four-Term Exact Sequences.- 4.1.8 Operations on Subgroups.- 4.1.9 p-Sylow Subgroups of Finite Abelian Groups.- 4.1.10 Enumeration of Subgroups.- 4.1.11 Application to the Solution of Linear Equations and Congruences.- 4.2 Computing the Structure of (?K/m) .- 4.2.1 Standard Reductions of the Problem.- 4.2.2 The Use of p-adic Logarithms.- 4.2.3 Computing (?K/pk) by Induction.- 4.2.4 Representation of Elements of (?K/m) .- 4.2.5 Computing (?K/m

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