The computation of invariants of algebraic number fields such as integral bases, discriminants, prime decompositions, ideal class groups, and unit groups is important both for its own sake and for its numerous applications, for example, to the solution of Diophantine equations. The practical com pletion of this task (sometimes known as the Dedekind program) has been one of the major achievements of computational number theory in the past ten years, thanks to the efforts of many people. Even though some practical problems still exist, one can consider the subject as solved in a satisfactory manner, and it is now routine to ask a specialized Computer Algebra Sys tem such as Kant/Kash, liDIA, Magma, or Pari/GP, to perform number field computations that would have been unfeasible only ten years ago. The (very numerous) algorithms used are essentially all described in A Course in Com putational Algebraic Number Theory, GTM 138, first published in 1993 (third corrected printing 1996), which is referred to here as [CohO]. That text also treats other subjects such as elliptic curves, factoring, and primality testing. Itis important and natural to generalize these algorithms. Several gener alizations can be considered, but the most important are certainly the gen eralizations to global function fields (finite extensions of the field of rational functions in one variable overa finite field) and to relative extensions ofnum ber fields. As in [CohO], in the present book we will consider number fields only and not deal at all with function fields.
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ISBN-13
978-1-4419-8489-0 (9781441984890)
DOI
10.1007/978-1-4419-8489-0
Schweitzer Klassifikation
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.
