Advanced Topics in Computational Number Theory

• 1. Fundamental Results and Algorithms in Dedekind Domains.- 1.1 Introduction.- 1.2 Finitely Generated Modules Over Dedekind Domains.- 1.3 Basic Algorithms in Dedekind Domains.- 1.4 The Hermite Normal Form Algorithm in Dedekind Domains.- 1.5 Applications of the HNF Algorithm.- 1.6 The Modular HNF Algorithm in Dedekind Domains.- 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.- 2.2 Arithmetic of Relative Extensions.- 2.3 Representation and Operations on Ideals.- 2.4 The Relative Round 2 Algorithm and Related Algorithms.- 2.5 Relative and Absolute Representations.- 2.6 Relative Quadratic Extensions and 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.- 3.3 Congruence Subgroups: One Side of Class Field Theory.- 3.4 Abelian Extensions: The Other Side of Class Field Theory.- 3.5 Putting Both Sides Together: The Takagi Existence Theorem 154.- 3.6 Exercises for Chapter 3.- 4. Computational Class Field Theory.- 4.1 Algorithms on Finite Abelian groups.- 4.2 Computing the Structure of (?K/m)*.- 4.3 Computing Ray Class Groups.- 4.4 Computations in Class Field Theory.- 4.5 Exercises for Chapter 4.- 5. Computing Defining Polynomials Using Kummer Theory.- 5.1 General Strategy for Using Kummer Theory.- 5.2 Kummer Theory Using Hecke’s Theorem When ?? ? K.- 5.3 Kummer Theory Using Hecke When ?? ? K.- 5.4 Explicit Use of the Artin Map in Kummer Theory When ?n ? K.- 5.5 Explicit Use of the Artin Map When ?n ? K.- 5.6 Two Detailed Examples.- 5.7 Exercises for Chapter 5.- 6. Computing Defining PolynomialsUsing Analytic Methods.- 6.1 The Use of Stark Units and Stark’s Conjecture.- 6.2 Algorithms for Real Class Fields of Real Quadratic Fields.- 6.3 The Use of Complex Multiplication.- 6.4 Exercises for Chapter 6.- 7. Variations on Class and Unit Groups.- 7.1 Relative Class Groups.- 7.2 Relative Units and Regulators.- 7.3 Algorithms for Computing Relative Class and Unit Groups.- 7.4 Inverting Prime Ideals.- 7.5 Solving Norm Equations.- 7.6 Exercises for Chapter 7.- 8. Cubic Number Fields.- 8.1 General Binary Forms.- 8.2 Binary Cubic Forms and Cubic Number Fields.- 8.3 Algorithmic Characterization of the Set U.- 8.4 The Davenport-Heilbronn Theorem.- 8.5 Real Cubic Fields.- 8.6 Complex Cubic Fields.- 8.7 Implementation and Results.- 8.8 Exercises for Chapter 8.- 9. Number Field Table Constructions.- 9.1 Introduction.- 9.2 Using Class Field Theory.- 9.3 Using the Geometry of Numbers.- 9.4 Construction of Tables of Quartic Fields.- 9.5 Miscellaneous Methods (in Brief).- 9.6 Exercises for Chapter 9.- 10. Appendix A: Theoretical Results.- 10.1 Ramification Groups and Applications.- 10.2 Kummer Theory.- 10.3 Dirichlet Series with Functional Equation.- 10.4 Exercises for Chapter 10.- 11. Appendix B: Electronic Information.- 11.1 General Computer Algebra Systems.- 11.2 Semi-general Computer Algebra Systems.- 11.3 More Specialized Packages and Programs.- 11.4 Specific Packages for Curves.- 11.5 Databases and Servers.- 11.6 Mailing Lists, Websites, and Newsgroups.- 11.7 Packages Not Directly Related to Number Theory.- 12. Appendix C: Tables.- 12.1 Hilbert Class Fields of Quadratic Fields.- 12.2 Small Discriminants.- Index of Notation.- Index of Algorithms.- General Index.

"Das vorliegende Buch ist eine Fortsetzung des bekannten erkes "A Course in Computational Algebraic Number Theory" (Graduate Texts in Mathematics 138) desselben Autors. ... So ist das vorliegende Buch ein sehr umfangliches Nachschlagewerk zur algorithmischen Zahlentheorie, das zusammen mit dem ersten Buch des Autors sicherlich eine Standard-Referenz fur zahlentheoretische Algorithmen darstellen wird." Internationale Mathematische Nachrichten, Nr. 187, August 2001