The valuation criterion for normal basis generators
type: Item
instance of: scholarly article
title: The valuation criterion for normal basis generators
zbMATH Open document ID: 1253.11108
author: Bart de Smit
author: Mathieu Florence
author: Lara Thomas
published in: Bulletin of the London Mathematical Society
publication date: 2012-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.2480
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 11S15
zbMATH DE Number: 6064195
zbMATH Keywords: local field
zbMATH Keywords: normal field basis
zbMATH Keywords: totally ramified
MaRDI profile type: Publication
OpenAlex ID: W3105575279
arXiv ID: 1004.2480
DOI: 10.1112/BLMS/BDS005
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review text: Let \(L/K\) be a finite Galois extension of local fields with Galois group \(G\), and let \(v\) denote the discrete valuation on \(L\). Starting with a result of \textit{N.~P. Byott} and \textit{G.~G. Elder} [Bull. Lond. Math. Soc. 39, No. 5, 705--708 (2007; Zbl 1128.11055)] one wants to characterize all such extensions which have the following property, called (VC): There exists some \(d \in \mathbb Z\) such that every element \(x \in L\) with \(v(x)=d\) is a normal generator, i.~e. the conjugates of \(x\) are a \(K\)-basis for \(L\).NEWLINENEWLINEIf (VC) holds, it is shown that \(L/K\) must be totally ramified and that for \(d\) there is only one possible value modulo \([L:K]\). From this the authors obtain the following criterion: (VC) holds for a non-trivial extension \(L/K\) iff the residue characteristic \(p\) is positive, \(L/K\) is totally ramified of degree some power of \(p\), and every non-zero \(K[G]\)-submodule of \(L\) contains some \(u\) with \(v(u)=0\).NEWLINENEWLINEAssuming that \(K\) is of mixed characteristic \((0,p)\), the authors investigate the latter condition for totally ramified Kummer extensions, and from this derive Theorem 1.4, which gives an explicit characterization of (VC) for abelian extensions of local fields.
reviewed by: Günter Lettl
description: scientific article; zbMATH DE number 6064195