Steinitz classes of Galois extensions with Galois group having nontrivial center
type: Item
instance of: scholarly article
title: Steinitz classes of Galois extensions with Galois group having nontrivial center
zbMATH Open document ID: 1311.11106
author: Bouchaïb Sodaïgui
published in: Journal of Number Theory
publication date: 2013-01-10T00:00:00Z
review text: Let \(K/k\) be an extension of number fields, let \(\mathcal O_K\) and \(\mathcal O_k\) be the rings of integers of \(K\) and \(k\) respectively. Then there exists an ideal \(I\) of \(\mathcal O_k\) and an isomorphism \(\mathcal O_K\cong \mathcal O_k^{[K:k]-1}\oplus I\) of \(\mathcal O_k\)-modules. The class \(\mathrm{cl}_k(I)\) of \(I\) in the ideal class group \(\mathrm{Cl}(k)\) of \(k\) is uniquely determined by \(K/k\) and is called its Steinitz class. Given a finite group \(\Gamma\), the set \(R_m(k,\Gamma)\) is defined as the subset of \(\mathrm{Cl}(k)\) of all the Steinitz classes of some tame Galois extensions of \(k\) with Galois group isomorphic to \(\Gamma\). It has been conjectured that \(R_m(k,\Gamma)\) is always a subgroup of \(\mathrm{Cl}(k)\). The conjecture is known to hold when \(\Gamma\) is abelian, as consequence of the main result of \textit{L. R. McCulloh} [J. Reine Angew. Math. 375/376, 259--306 (1987; Zbl 0619.12008)], and for several non-abelian groups. The main difficulty when \(\Gamma\) is non-abelian comes from the fact that one needs to construct some Galois extensions of \(k\) with Galois group isomorphic to \(\Gamma\) and with some given Steinitz class. In the paper under review the author develops a different approach to avoid this kind of problem: he assumes the existence of a tame extension of \(k\) with Galois group \(\Gamma\) and he modifies this extension to obtain some new extensions with the same Galois group, but with different Steinitz classes. In general the Steinitz classes realized in this way do not form a group and should not be expected to be the whole of \(R_m(k,\Gamma)\), but the author gives some conditions under which they coincide with the whole ideal class group of \(k\), which implies that \(R_m(k,\Gamma)=\mathrm{Cl}(k)\). More concretely, the requests are that the center \(Z(\Gamma)\) of \(\Gamma\) is non-trivial and the class number of \(k\) is coprime to an integer \(n_p\), which depends on a prime \(p\) divid…
reviewed by: Alessandro Cobbe
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 6124185
zbMATH Keywords: Galois module structure
zbMATH Keywords: ring of integers
zbMATH Keywords: realizable Steinitz classes
zbMATH Keywords: central extensions
zbMATH Keywords: solvable groups
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.08.015
OpenAlex ID: W2028421994
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DOI: 10.1016/J.JNT.2012.08.015
description: scientific article; zbMATH DE number 6124185