Advanced Topics in Computational Number Theory

1. Front Matter

   Pages i-xv

   PDF

2. Fundamental Results and Algorithms in Dedekind Domains

   • Henri Cohen

   Pages 1-47

3. Basic Relative Number Field Algorithms

   • Henri Cohen

   Pages 49-132

4. The Fundamental Theorems of Global Class Field Theory

   • Henri Cohen

   Pages 133-162

5. Computational Class Field Theory

   • Henri Cohen

   Pages 163-222

6. Computing Defining Polynomials Using Kummer Theory

   • Henri Cohen

   Pages 223-295

7. Computing Defining Polynomials Using Analytic Methods

   • Henri Cohen

   Pages 297-346

8. Variations on Class and Unit Groups

   • Henri Cohen

   Pages 347-387

9. Cubic Number Fields

   • Henri Cohen

   Pages 389-428

10. Number Field Table Constructions

    • Henri Cohen

    Pages 429-473

11. Appendix A: Theoretical Results

    • Henri Cohen

    Pages 475-521

12. Appendix B: Electronic Information

    • Henri Cohen

    Pages 523-531

13. Appendix C: Tables

    • Henri Cohen

    Pages 533-548

14. Back Matter

    Pages 549-581

    PDF

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.

• Euclidean algorithm
• Multiplication
• Prolog
• algorithms
• arithmetic
• boundary element method
• computation
• construction
• database
• eXist
• field
• function
• functional equation
• number theory
• prime number
• combinatorics

• Lab. Algorithmique Arithmétique Expérimentale, Université de Bordeaux 1, Talence, France

  Henri Cohen