On the Stickelberger ideal of (2,\dots, 2)-extensions of a cyclotomic number field
type: Item
instance of: scholarly article
title: On the Stickelberger ideal of (2,\dots, 2)-extensions of a cyclotomic number field
zbMATH Open document ID: 0715.11061
DOI: 10.1007/BF02567915
author: Akira Endo
published in: Manuscripta Mathematica
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155548
review text: For rational integers \(m_1,\ldots,m_t\), let \(K\) denote the abelian field generated by \(\sqrt{m_1},\ldots,\sqrt{m_t}\) over the cyclotomic field of \(p^n\)-th roots of 1 \((p\) an odd prime, \(n\ge 1)\). The author constructs a system of generators for the minus part of the Stickelberger ideal \(S\) of \(\mathbb Z[G]\), where \(G=\mathrm{Gal}(K/\mathbb Q)\). This gives an expression for the index \((\mathbb Z[G]^-:S^-)\) in terms of certain determinants \(D_{p^n}\) with elements in \(\mathbb Z\). Upon calculating these determinants (each of which depends on a character \(\chi\) of \(K)\) one arrives at a formula connecting this index to the relative class number of \(K\). The formula in fact is a special case of one proved by \textit{C.-G. Schmidt} [J. Number Theory 11, No. 1, 128--159 (1979; Zbl 0403.12009)] for arbitrary abelian fields. Finally, the author finds a natural relationship between the \(p\)-divisibility of \(D_p\) and the \(\chi\)- irregularity of \(p\).
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 4177202
zbMATH Keywords: abelian field
zbMATH Keywords: cyclotomic field
zbMATH Keywords: Stickelberger ideal
zbMATH Keywords: relative class number
MaRDI profile type: Publication
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OpenAlex ID: W1964496400
description: scientific article; zbMATH DE number 4177202