An index formula for the relative class number of an abelian number field
type: Item
instance of: scholarly article
title: An index formula for the relative class number of an abelian number field
zbMATH Open document ID: 0675.12002
DOI: 10.1016/0022-314X(89)90100-5
author: Kurt Girstmair
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: It is shown that the relative class number \(h^-_ K\), of an imaginary abelian field K (of \(conductor\quad n\) and ring of \(integers\quad {\mathfrak O}),\) is given by: \((*)\quad c_ Kh^-_ K=((1-\tau){\mathfrak O}:<\eta >),\) where \(<\eta >\) is the additive \({\mathbb{Z}}[Gal(K/{\mathbb{Q}})]\)-module generated by the trace \(\eta\), in \({\mathbb{Q}}(\zeta_ n)/K\), of \(i\cdot \cot g(\pi /n)=(1+\zeta_ n)/1-\zeta_ n)\) \((\zeta_ n=\exp (2i\pi /n)\), \(\tau =complex\) conjugation); here the rational \(c_ K\neq 0\) depends on standard Euler factors and elementary invariants of \(K/{\mathbb{Q}}.\) The computation of \(((1-\tau){\mathfrak O}:<\eta>)\) involves Leopoldt's ``character coordinates'' [\textit{H. W. Leopoldt}, J. Reine Angew. Math. 201, 119-149 (1959; Zbl 0098.034)] for odd character \(\chi\); for \(\eta\), these coordinates are essentially Bernoulli numbers \(B_{1,\chi}\), and the result follows from a classical analytic formula of \(h^-_ K\) (Kubert-Lang formulas are also described with similar indices).
reviewed by: Georges Gras
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 4105076
zbMATH Keywords: relative class number
zbMATH Keywords: imaginary abelian field
zbMATH Keywords: Kubert-Lang formulas
MaRDI profile type: Publication
cites work: Ein v. Staudt-Clausenscher Satz für periodische Bernoullizahlen. (A theorem of von Staudt-Clausen type for periodic Bernoulli numbers)
cites work: Character coordinates and annihilators of cyclotomic numbers
cites work: Q5843509
cites work: A class number formula for cyclotomic fields
cites work: Stickelberger ideals
cites work: Q5832112
cites work: Eine Verallgemeinerung der Bernoullischen Zahlen
cites work: Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers.
cites work: Größencharaktere und Relativklassenzahl abelscher Zahlkörper
cites work: On the Stickelberger ideal and the circular units of a cyclotomic field
cites work: On the Stickelberger ideal and the circular units of an abelian field
cites work: Q3943047
description: scientific article; zbMATH DE number 4105076