Author | Cohen, Henri. author |
---|---|
Title | Advanced Topics in Computational Number Theory [electronic resource] / by Henri Cohen |
Imprint | New York, NY : Springer New York : Imprint: Springer, 2000 |
Connect to | http://dx.doi.org/10.1007/978-1-4419-8489-0 |
Descript | XV, 581 p. online resource |
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 Heckeโ{128}{153}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 Polynomials Using Analytic Methods -- 6.1 The Use of Stark Units and Starkโ{128}{153}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