Realizable Galois module classes over the group ring for non abelian extensions
type: Item
instance of: scholarly article
title: Realizable Galois module classes over the group ring for non abelian extensions
zbMATH Open document ID: 1316.11105
author: Nigel P. Byott
author: Bouchaïb Sodaïgui
publication date: 2013-06-19T00:00:00Z
full work available at URL: http://hdl.handle.net/10871/11745
review text: We say that an extension of number fields is tame if it is at most tamely ramified. Fix a number field \(k\) and a finite group \(\Gamma\). We say that \(N\) is a \(\Gamma\)-extension of \(k\) if \(N/k\) is a Galois extension of number fields equipped with an isomorphism \(\mathrm{Gal}(N/k) \rightarrow \Gamma\). If \(N\) is a tame \(\Gamma\)-extension of \(k\) then the ring of integers \(\mathcal{O}_{N}\) is a locally free module of rank \(1\) over the group ring \(\mathcal{O}_{k}[\Gamma]\), and its structure as such is determined up to stable isomorphism by its class \((\mathcal{O}_{N})_{\mathcal{O}_{k}[\Gamma]}\) in the locally free class group \(\mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\). The set \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) of realizable classes is defined to be the subset of \(\mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\) consisting of the classes \((\mathcal{O}_{N})_{\mathcal{O}_{k}[\Gamma]}\) as \(N\) runs through all tame \(\Gamma\)-extensions of \(k\). It is a conjecture that \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) is in fact a subgroup of \(\mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\). This has been proven by \textit{L. R. McCulloh} in the case that \(\Gamma\) is abelian see [J. Reine Angew. Math. 375/376, 259--306 (1987; Zbl 0619.12008)]. However, much less is known in the non-abelian case; indeed, a proof that \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) is non-empty would solve the inverse Galois problem for \(\Gamma\) over \(k\). The article under review determines \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) for certain metabelian groups under a mild restriction on \(k\). In particular, the results show that \(\mathcal{R}(\mathcal{O}_{k}[\Gamma])\) is indeed a subgroup of \(\mathrm{Cl}(\mathcal{O}_{k}[\Gamma])\) in the cases under consideration. Let \(p\) be a prime and assume that \(k\) contains a primitive \(p\)th root of unity. Let \(\Gamma\) be the semidirect product \(V \rtimes C\) of an elementary abelian group \(V\) of order \(p^{r}\) by any cyclic group \…
reviewed by: Henri Johnston
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 6177083
zbMATH Keywords: Galois module structure
zbMATH Keywords: rings of algebraic integers
zbMATH Keywords: locally free class group
zbMATH Keywords: Fröhlich-Lagrange resolvent
zbMATH Keywords: realizable classes
zbMATH Keywords: embedding problem
zbMATH Keywords: Stickelberger ideal
zbMATH Keywords: cyclic codes
MaRDI profile type: Publication
OpenAlex ID: W605691493
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DOI: 10.5802/AIF.2762
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 6177083