Generalizations of Hensel's lemma and the nearest root method
type: Item
instance of: scholarly article
title: Generalizations of Hensel's lemma and the nearest root method
zbMATH Open document ID: 1270.12001
published in: Algebra and Logic
publication date: 2012-12-17T00:00:00Z
review text: Let \(v\) be a Henselian valuation of arbitrary rank of a field \(K\) and \(\tilde{v}\) be the prolongation of \(v\) to the algebraic closure \(\widetilde{K}\) of \(K\) with value group \(\widetilde{G}\). In 2008, \textit{Ron Brown} [Commun. Algebra 37, No. 7, 2169--2183 (2009; Zbl 1205.12006)] gave a class \(\mathcal{P}\) of monic irreducible polynomials over \(K\) such that to each \(g(x)\) belonging to \(\mathcal{P}\), there corresponds a smallest constant \(\lambda_g\) belonging to \(\widetilde{G}\) with the property that whenever \(\tilde{v}(g(\beta))\) is more than \(\lambda_g\) with \(K(\beta)\) a tamely ramified extension of \((K,v)\), then \(K(\beta)\) contains a root of \(g(x)\). In 2010, a method was described in [the reviewer, J. Pure Appl. Algebra 214, No. 12, 2294--2300 (2010; Zbl 1244.12002)], to explicitly determine this constant besides giving its significant properties. In the present article the author proves a result leading to a new proof of these properties.
reviewed by: Sudesh Kaur Khanduja
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12J25
zbMATH DE Number: 6116203
zbMATH Keywords: irreducible polynomials
zbMATH Keywords: valued fields
zbMATH Keywords: non-archimedean valued fields
author: Yuri L. Ershov
MaRDI profile type: Publication
full work available at URL: s10469-012-9159-8
OpenAlex ID: W2027579152
Wikidata QID: Q125060990
cites work: New light on Hensel's Lemma
cites work: On Brown's constant associated with irreducible polynomials over Henselian valued fields
cites work: Lubin-Tate extensions, an elementary approach
cites work: Q5437086
cites work: INVARIANTS OF DEFECTLESS IRREDUCIBLE POLYNOMIALS
cites work: On Chains Associated with Elements Algebraic over a Henselian Valued Field
cites work: On Krasner's constant
DOI: 10.1007/S10469-012-9159-8
description: scientific article; zbMATH DE number 6116203