Realizable classes of metacylic extensions of degree \(lm\)
type: Item
instance of: scholarly article
title: Realizable classes of metacylic extensions of degree \(lm\)
zbMATH Open document ID: 1200.11085
author: Farah Sbeity
author: Bouchaïb Sodaïgui
published in: Journal of Number Theory
publication date: 2010-07-07T00:00:00Z
review text: Let \(O_k\) be the ring of integers of some number field \(k\), and \(\Gamma\) a metacyclic group of order \(lm\), where \(l\) is a rational prime and \(m\) some positive integer. Let \(\mathcal M\) be a maximal \(O_k\)-order in \(k[\Gamma]\), which contains \(O_k[\Gamma]\), and \(Cl (\mathcal M)\) the class group of locally free \(\mathcal M\)-modules. One is interested to study the subset \(\mathcal R (\mathcal M) \subset Cl (\mathcal M)\) of realisable classes, i.e. of those classes \(c\) such that there exists some tame Galois extension \(N/k\) with Galois group isomorphic to \(\Gamma\) and with \(\mathcal M \otimes_{O_k [\Gamma]} O_N\) lying in \(c\). It is conjectured that \(\mathcal R (\mathcal M)\) is even a subgroup of \(Cl (\mathcal M)\), which is true in the abelian case. In the present paper, the authors investigate the subset \(\mathcal R_1 (\mathcal M) \subset \mathcal R (\mathcal M)\) arising from those extensions \(N/k\) which are linearly disjoint from \(k(\zeta)/k\), where \(\zeta\) denotes an \(l\)-th root of unity. They use a Stickelberger ideal to give an explicit description of \(\mathcal R_1 (\mathcal M)\) and show that \(\mathcal R_1 (\mathcal M)\) is indeed a group.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 5731637
zbMATH Keywords: Galois module structure
zbMATH Keywords: rings of integers
zbMATH Keywords: locally free class group
zbMATH Keywords: Stickelberger ideal
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.10.007
OpenAlex ID: W1991330793
cites work: Q5798080
cites work: On realizable Galois module classes and Steinitz classes of nonabelian extensions
cites work: Classes réalisables d'extensions non abéliennes
cites work: Realizable Galois module classes for tetrahedral extensions
cites work: Galois module structure for dihedral extensions of degree 8: realizable classes over the group ring
cites work: Classes de Steinitz d'extensions quaternioniennes généralisées de degré 4p r
cites work: Q3215048
cites work: Q4724810
cites work: The discriminants of relative extensions and the existence of integral bases
cites work: Arithmetic and Galois module structure for tame extensions.
cites work: Q4154965
cites work: Q3966247
cites work: Q4739993
cites work: Sur les corps résolubles de degré premier.
cites work: Q4170825
cites work: Galois module structure of abelian extensions.
cites work: Critère de décomposition d'une extension de Kummer sur un sous-corps du corps de base
cites work: Q4414310
cites work: Relative Galois structure of rings of integers.
cites work: Realizable classes by non-abelian metacyclic extensions and Stickelberger elements
cites work: Realizable classes of quaternion extensions of degree \(4\ell\)
cites work: Relative Galois module structure of octahedral extensions
cites work: Steinitz Classes of Metacyclic Extensions
cites work: On Fröhlich's conjecture for rings of integers of tame extensions
DOI: 10.1016/J.JNT.2009.10.007
description: scientific article; zbMATH DE number 5731637