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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Efficient computation of root numbers and class numbers of parametrized families of real abelian number fields


Author: Stéphane R. Louboutin
Journal: Math. Comp. 76 (2007), 455-473
MSC (2000): Primary 11R16, 11R20, 11R29, 11R42, 11Y40
Published electronically: September 11, 2006
MathSciNet review: 2261031
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Abstract: Let $ \{K_m\}$ be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials $ P_m(x) =x^3 -mx^2-(m+3)x+1$ and $ P_m(x) =x^4 -mx^3-6x^2+mx+1$. We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest number fields. We deduce a method for computing the exact values of these Gaussian sums. These values are then used to efficiently compute class numbers of simplest fields. Finally, such class number computations yield many examples of real cyclotomic fields $ {\bf Q}(\zeta_p)^+$ of prime conductors $ p\ge 3$ and class numbers $ h_p^+$ greater than or equal to $ p$. However, in accordance with Vandiver's conjecture, we found no example of $ p$ for which $ p$ divides $ h_p^+$.


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Additional Information

Stéphane R. Louboutin
Affiliation: Institut de Mathématiques de Luminy, UMR 6206, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
Email: loubouti@iml.univ-mrs.fr

DOI: http://dx.doi.org/10.1090/S0025-5718-06-01888-6
PII: S 0025-5718(06)01888-6
Keywords: Real abelian number field, class number, Gauss sums, simplest cubic field, simplest quartic field, simplest quintic field, simplest sextic field.
Received by editor(s): July 8, 2005
Received by editor(s) in revised form: October 14, 2005
Published electronically: September 11, 2006
Dedicated: Dedicated to Danièle B.
Article copyright: © Copyright 2006 American Mathematical Society