The relative class number of certain imaginary abelian fields
type: Item
instance of: scholarly article
title: The relative class number of certain imaginary abelian fields
zbMATH Open document ID: 0699.12015
DOI: 10.1007/BF02941380
author: N. E. Zubov
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1988-01-01T00:00:00Z
review text: Let \(m\neq 1\) be a square-free integer, prime to an odd prime \(p\), and let \(K=\mathbb Q(\sqrt{m},\zeta)\) where \(\zeta\) denotes a primitive \(p^ n\)-th root of unity (\(n\) a positive integer). The author considers the minus parts \(\mathbb Z[G]^-\), and \(S^-\) of the group ring \(\mathbb Z[G]\) \((G=\text{Gal}(K/\mathbb Q))\), and the Stickelberger ideal \(S\) of \(\mathbb Z[G]\). He mentions a special system of generators of \(S^-\) considered as a \(\mathbb Z\)-module, and he expresses the group index \((\mathbb Z[G]^- : S^-)\) by means of a product of two special determinants. Using a formula of \textit{K. Iimura} [Arch. Math. 36, 45--52 (1981; Zbl 0465.12002)] he obtains a new formula for the relative class number \(h^-_ K\) of \(K\).
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 4146056
zbMATH Keywords: quadratic extension of cyclotomic field
zbMATH Keywords: Stickelberger ideal
zbMATH Keywords: generators
zbMATH Keywords: group index
zbMATH Keywords: relative class number
reviewed by: Ladislav Skula
MaRDI profile type: Publication
cites work: Q5521595
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description: scientific article; zbMATH DE number 4146056