Realizable classes by non-abelian metacyclic extensions and Stickelberger elements
type: Item
instance of: scholarly article
title: Realizable classes by non-abelian metacyclic extensions and Stickelberger elements
zbMATH Open document ID: 0873.11061
author: Bouchaïb Sodaïgui
published in: Journal of Number Theory
publication date: 1997-07-24T00:00:00Z
review text: A theorem of McCulloh (part of which was shown earlier by Childs) states that for a group \(\Gamma\) of prime order and a number field \(k\), the classes in \(Cl({\mathcal O}_k[\Gamma])\) arising as \([{\mathcal O}_N]\) with \(N/k\) a tame \(\Gamma\)-extension are exactly those which are in \(J\cdot Cl({\mathcal O}_k[\Gamma])\), where \(J\subset {\mathbb{Z}}[Aut(\Gamma)]\) is a suitably defined Stickelberger ideal. The paper under review studies the simplest case of a non-commutative Galois group: \(\Gamma\) is now supposed to be metacyclic non-abelian of order \(lq\), where \(l\) and \(q\) are prime numbers and \(q\) divides \(l-1\). The author proves a very nice result which determines the classes which are realizable by tame \(\Gamma\)-extensions \(N/k\), with the following change of scenario: one works in the class group of a maximal order \(\mathcal M\) in \(k[\Gamma]\), instead of the class group of \({\mathcal O}_k[\Gamma]\) itself. The result extends McCulloh's result in a satisfactory way: again Stickelberger ideals are the clue to the solution. One can get started since the class group of the above mentioned maximal order is easily obtained by Morita theory once one knows the decomposition of the group algebra \(k[\Gamma]\) into simple components. In fact, \(Cl({\mathcal M})\) is the product of the three class groups \(Cl({\mathcal O}_k)\), \(Cl({\mathcal O}_{k(\zeta_q)})\), and \(Cl({\mathcal O}_K)\), where \(K\) is the unique field between \(k\) and \(k(\zeta_l)\) such that \([k(\zeta_l):K]=q\). The proof of the main result uses the results for the abelian prime order case in a neat and efficient way. This is a line of research which deserves to be continued.
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1037293
zbMATH Keywords: Stickelberger ideals
zbMATH Keywords: Galois modules
zbMATH Keywords: realizable classes
zbMATH Keywords: non-abelian metacyclic extensions
zbMATH Keywords: Stickelberger elements
MaRDI profile type: Publication
full work available at URL: jnth.1997.2133
OpenAlex ID: W2063380619
cites work: Q3966247
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cites work: Q5628326
cites work: Relative Galois structure of rings of integers.
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DOI: 10.1006/JNTH.1997.2133
description: scientific article; zbMATH DE number 1037293