Computation of class numbers of number fields
type: Item
instance of: scholarly article
title: Computation of class numbers of number fields
zbMATH Open document ID: 0854.11060
author: Stéphane Louboutin
published in: Pacific Journal of Mathematics
publication date: 1996-11-07T00:00:00Z
review text: Let \(K/k\) be an extension of number fields. Assume that the quotient \(\zeta_K/ \zeta_k\) of the Dedekind zeta-function of \(K\) and that of \(k\) is homomorphic in the whole complex plane. Note that this assumption holds as soon as \(K/k\) is normal. We give an expression for the value \((\zeta_K/ \zeta_k) (1)\) as the limit of a rapidly convergent series, which makes it convenient to numerically compute this value \((\zeta_K/ \zeta_k) (1)\). Our main purpose is to compute relative class numbers of non-abelian CM-fields \(K\), taking \(k\) as the maximal totally real subfield of \(K\) (here \(K/k\) is quadratic, thus normal). For example, we explain in detail how to compute the relative class number of a non-normal quartic CM-field, and provide tables of relative class numbers of such CM-fields. The reader may find other illustrations by example of the method developed in this paper in the following two papers [`Calcul des nombres de classes relatifs: applications aux corps octiques quaternioniques à multiplication complexe', C. R. Acad. Sci., Paris, Sér. I 317, 643-646 (1993; Zbl 0795.11059), and `Calcul des nombres de classes relatifs de certains corps de classes de Hilbert', ibid. 319, 321-325 (1994; Zbl 0817.11049)].
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 912155
zbMATH Keywords: Dedekind zeta-function
zbMATH Keywords: relative class numbers
zbMATH Keywords: non-abelian CM-fields
zbMATH Keywords: non-normal quartic CM-field
zbMATH Keywords: tables
MaRDI profile type: Publication
full work available at URL: pjm.1995.171.455
OpenAlex ID: W1564589490
DOI: 10.2140/PJM.1995.171.455
description: scientific article; zbMATH DE number 912155