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Efficient Computation of Class Numbers of Real Abelian Number Fields

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Algorithmic Number Theory (ANTS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2369))

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Abstract

Let K m be a parametrized family of real abelian number fields of known regulators, e.g. the simplest cubic fields associated with the Q-irreducible cubic polynomials P m (x) = x 3 − mx 2 − (m + 3)x − 1. We develop two methods for computing the class numbers of these Km’s. As a byproduct of our computation, we found 32 cyclotomic fields Q(ζp) of prime conductors p < 1010 for which some prime q ≥ p divides the class numbers h + p of their maximal real subfields Q(ζp)+ (but we did Not find any conterexample to Vandiver’s conjecture!).

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Author information

Authors and Affiliations

  1. Institut de Mathématiques de Luminy, UPR 906 163, avenue de Luminy, Case 907, 13288, Marseille Cedex 9, France

    Stéphane R. Louboutin

Authors
  1. Stéphane R. Louboutin
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Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW, 2006, Australia

    Claus Fieker  & David R. Kohel  & 

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Louboutin, S.R. (2002). Efficient Computation of Class Numbers of Real Abelian Number Fields. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_11

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  • DOI: https://doi.org/10.1007/3-540-45455-1_11

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43863-2

  • Online ISBN: 978-3-540-45455-7

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