Tame and Galois extensions with respect to Hopf orders
type: Item
instance of: scholarly article
title: Tame and Galois extensions with respect to Hopf orders
zbMATH Open document ID: 0841.16021
DOI: 10.1007/BF02572628
author: Nigel P. Byott
published in: Mathematische Zeitschrift
publication date: 1996-07-17T00:00:00Z
full work available at URL: https://eudml.org/doc/174823
review text: Let \(K\) be the field of fractions of a Dedekind domain \(R\), let \(G\) be a finite group, let \(\mathfrak G\) be a Hopf order in \(KG\), and let \(S\) be an order admitting \(\mathfrak G\) in a Galois algebra \(L\) over \(K\) with group \(G\). We consider when \(S\) is a Galois or tame \(\mathfrak G\)-extension of \(R\) in the sense of L. N. Childs. We give a criterion for \(S\) to be a Galois \(\mathfrak G\)-extension, involving the inverse differents of various subextensions. We then reduce the study of tame and Galois extensions to a special case, showing that if \(S\) is an integrally closed extension of \(R\), and the corresponding extension of residue class fields is separable, then we may take \(S\) to be a complete discrete valuation ring and \(G\) to be a \(p\)-group. In this case, we give a criterion on the ramification numbers of \(L/K\) for \(S\) to be a Galois \(\mathfrak G\)-extension, and show that if \(S\) is a tame \(\mathfrak G\)-extension then \(S\) is a Galois \(\mathfrak G\)-extension, unless \(G\) is cyclic and \(\mathfrak G\) is the maximal order in \(KG\).
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 11S45
zbMATH DE Number: 856578
zbMATH Keywords: field of fractions
zbMATH Keywords: finite groups
zbMATH Keywords: Hopf orders
zbMATH Keywords: tame \(\mathfrak G\)-extensions
zbMATH Keywords: Galois extensions
zbMATH Keywords: integrally closed extensions
zbMATH Keywords: ramification numbers
zbMATH Keywords: Galois \(\mathfrak G\)-extensions
zbMATH Keywords: maximal orders
MaRDI profile type: Publication
Wikidata QID: Q57920063
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OpenAlex ID: W2076641455
description: scientific article; zbMATH DE number 856578