Skip to main content
SpringerLink
Log in

From Class Groups to Class Fields

  • Conference paper
Algorithmic Algebra and Number Theory

Abstract

Since the first printing of the book [35] by H. Zassenhaus and the author in 1989 algorithmic algebraic number theory has attracted rapidly increasing interest. This is documented, for example, by a regular meeting ANTS (algebraic number theory symposium) every two years whose proceedings [1], [6] give a good survey about ongoing research. Also there are several computer algebra packages concentrating on number theoretical computations. At present the most prominent ones, which are available for free, are Kant [12], Pari [2] and Simath [27]. Kant comes with a data base for algebraic number fields, already containing more than a million fields of small degree. Kant is developed by the research group of the author at Berlin and will be presented in some detail in section 5. We note that almost all of Kant and Pari is also contained in the Magma system [4].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L.M. Adleman, M.-D. Huang (Eds.), Algorithmic Number Theory Symposium, Proceedings ANTS I, Cornell University 1994.

    Google Scholar 

  2. D. Bernardi, C. Batut, H. Cohen, M. Olivier, User’s guide to PARI-GP, version 1.39, Publ. Université Bordeaux I, 1991.

    Google Scholar 

  3. W. Bosma and M. Pohst, Computations with finitely generated modules over Dedekind rings,Proc. ISSAC’91(1991), 151–156.

    Google Scholar 

  4. W. Bosma, J.J. Cannon and C. Playoust, The Magma Algebra System I: The User Language, J. Symb.Comp. 24 (1997), 235–265.

    MathSciNet  MATH  Google Scholar 

  5. J. Buchmann, M. Jüntgen and M. Pohst, A Practical Version of the Generalized Lagrange Algorithm, Exper. Math. 3 (1994), 199–207.

    Google Scholar 

  6. H. Cohen (Ed.), Algorithmic Number Theory, Proc. Sec. Int. Symp., ANTS II, Bordeaux 1996, Lect. Notes in Compo Sc. 1122, Springer Verlag, Berlin-HeidelbergNew York 1996.

    Google Scholar 

  7. H. Cohen, Hermite and Smith normal form algorithms over Dedekind domains, to appear in Math. Comp ..

    Google Scholar 

  8. H. Cohen, F. Diaz y Diaz, and M. Olivier, Computing ray class groups, conductors and discriminants, submitted to Math. Comp ..

    Google Scholar 

  9. M. Daberkow, Öber die Bestimmung der ganzen Elemente in Radikalerweiterungen algebraischer Zahlkorper, Thesis, TU Berlin, 1995.

    Google Scholar 

  10. M. Daberkow and M. Pohst,On Integral Bases in Relative Quadratic Extensions, Math. Compo 65 (1996), 319–329.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Daberkow and M. Pohst, Computations with relative extensions of number fields with an application to the construction of Hilbert class fields, Proc. ISSAC’95, ACM Press, New York 1995, pp. 68–76.

    Google Scholar 

  12. M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symb. Compo 24(1997), 267–283.

    Article  MATH  Google Scholar 

  13. M. Daberkow and M. Pohst, On the computation of Hilbert class fields, to appear in J. Number Theory.

    Google Scholar 

  14. J. Dixon,Computing Subfields in Algebraic Number Fields, J. Austral. Math. Soc. (Series A) 49 (1990), 434–448.

    Article  MATH  Google Scholar 

  15. Y. Eichenlaub and M. Olivier, Computation of Galois groups for polynomials with degree up to eleven, to appear.

    Google Scholar 

  16. C. Fieker, Uber relative Normgleichungen in algebraischen Zahlkorpern, Thesis, TU Berlin, 1997.

    Google Scholar 

  17. C. Fieker, Computing class fields via the Artin map, preprint, Berlin, 1997.

    Google Scholar 

  18. C. Friedrichs, Berechnung relativer Ganzheitsbasen mit dem Round-~-Algorithmus, Diplomarbeit, TU Berlin, 1997.

    Google Scholar 

  19. I. Gaál, A. Pethöund M. Pohst,Simultaneous representation of integers by a pair of ternary quadratic forms - with an application to index form equations in quartic number fields, J. Number Theory 57(1996), 90–104.

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Gaál and M. Pohst, Power integral bases in a parametric family of totally real cyclic quintics, to appear in Math. Comp ..

    Google Scholar 

  21. I. Gaál and M. Pohst, On the resolution of index form equations in sextic fields with an imaginary quadratic subfield, to appear in J. Symb. Comp ..

    Google Scholar 

  22. J. Gebel, A. Pethöand H. G. Zimmer, On Mornell’s equation, to appear in Compos. Math ..

    Google Scholar 

  23. K. Geißler, Zur Berechnung von Galoisgruppen, Diplomarbeit, TU Berlin, 1997.

    Google Scholar 

  24. H. Hasse, Uber den Klassenkorper zum quadratischen Zahlkorper mit der Diskrimin ante -47, Acta Arithmetica 9 (1964), 419 – 434.

    MathSciNet  Google Scholar 

  25. H. Hasse, Vorlesungen iiber Klassenkorpertheorie, Physika Verläg, Würzburg, 1967.

    Google Scholar 

  26. F. Heβ, Zur Klassengruppenberechnung in algebraischen Zahlkörpern, Diplomarbeit, TU Berlin, 1996.

    Google Scholar 

  27. C. Hollinger and P. Serf, SIMATH - a computer algebra system, in Computational Number Theory, eds. A. Petho, M.E. Pohst, H.C. Williams, H.G. Zimmer, de Gruyter Verlag, Berlin, New York 1991, pp.331–342.

    Chapter  Google Scholar 

  28. G. J. Janusz, Algebraic Number Fields, 2nd ed., AMS Graduate Studies in Math., Vol. 7, 1996.

    MATH  Google Scholar 

  29. A. Jurk, Öber die Berechnung von Losungen relativer Normgleichungen in algebraischen Zahlkorpern, Dissertation, Düsseldorf, 1993.

    Google Scholar 

  30. J.Klüners, Öber die Berechnung von Automorphismen und Teilkorpern algebraischer Zahlkorper, Thesis, TUBerlin1997.

    Google Scholar 

  31. J. Klüners and M. Pohst, On Computing subfields, J. Symb. Compo 24(1997), 385–397.

    Article  MATH  Google Scholar 

  32. S. Lang, Algebraic number theory, Grad. Texts in Math. 110, Springer Verlag, 1994.

    MATH  Google Scholar 

  33. S.Pauli, Zur Berechnung von Strahlklassengruppen, Diplomarbeit, TUBerlin, 1996.

    Google Scholar 

  34. M. Pohst, On computing fundamental units, J. Number Theory 47 (1994), 93–105.

    Google Scholar 

  35. J. von Schmettow, Beitrage zur Klassengruppenberechnung, Dissertation, Diisseldorf 1991.

    Google Scholar 

  36. R. Schertz, Zur expliziten Berechnung von Ganzheitsbasen in Strahlklassenkorpern iiber imaginar-quadratischen Zahlkorpern, J. Number Theory 34(1990), 41–53.

    Article  MathSciNet  MATH  Google Scholar 

  37. M.Schörnig, Untersuchungen konstruktiver Probleme in globalen Funktionenkorpern, Thesis, TUBerlin, 1996.

    Google Scholar 

  38. J. Sommer, Einfiihrung in Zahlentheorie, Teubner Verlag, Leipzig 1907.

    Google Scholar 

  39. A. Weber, M. Daberkow,A Database for Number Fields, J. Calmet and C. Limongelli (Eds.), Design and Implementation of Symbolic Computation Systems,pp. 320–330, Springer LNCS 1128, Berlin - Heidelberg - New York, 1996.

    Google Scholar 

  40. K.Wildanger, Uber das Losen von Einheiten- und Indexformgleichungen in algebraischen Zahlkorpern mit einer Anwendung auf die Bestimmung aller ganzen Punkte einer Mordellschen Kurve, Thesis, TUBerlin, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich 3 Mathematik, Technische Universität Berlin, Ma 8-1, Strafie des 17. Juni 136, 10623, Berlin, Germany

    Michael E. Pohst

Authors
  1. Michael E. Pohst
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, der Universität Heidelberg, Im Neuenheimer Feld 368, D-69120, Heidelberg, Germany

    B. Heinrich Matzat

  2. Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    Gert-Martin Greuel

  3. Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, D-52062, Aachen, Germany

    Gerhard Hiss

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pohst, M.E. (1999). From Class Groups to Class Fields. In: Matzat, B.H., Greuel, GM., Hiss, G. (eds) Algorithmic Algebra and Number Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59932-3_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-59932-3_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64670-9

  • Online ISBN: 978-3-642-59932-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation