On Brown's constant associated with irreducible polynomials over Henselian valued fields
type: Item
instance of: scholarly article
title: On Brown's constant associated with irreducible polynomials over Henselian valued fields
zbMATH Open document ID: 1244.12002
published in: Journal of Pure and Applied Algebra
publication date: 2010-08-27T00:00:00Z
review text: Let \(v\) be a Henselian valuation of an arbitrary rank of a field \(K\) and \(\tilde v\) is the unique prolongation of \(v\) to the algebraic closure \(\tilde K\) of \(K\) with value group \(\tilde G\). In 2008, Brown gave a class of monic irreducible polynomials over \(K\) satisfying the following property: To every \(g(X)\in{\mathcal P}\), one can associate a constant \(\lambda_g\in\tilde G\) such that whenever \(K(\beta)\) is a tamely ramified extension of \((K,v)\), \(\beta\in\tilde K\) and \(\tilde v(g(\beta))>\lambda_g\), then \(K(\beta)\) contains a root of the polynomial \(g(X)\). Moreover, the constant \(\lambda_g\) is the smallest with the above property. This constant is referred to as Brown's constant. In this paper, the author determines explicitly Brown's constant for all possible irreducible polynomials \(g(X)\) and shows that this constant satisfies an important property even without the assumption that \(K(\beta)/K\) is tamely ramified. She shows that this constant can be associated to any monic irreducible polynomial \(g(X)\in K[X]\) provided \(K(\theta)\) is a defectless extension of \((K,v)\) where \(\theta\) is a root of \(g(X)\).
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12J25
zbMATH DE Number: 5775718
zbMATH Keywords: Henselian valuation
zbMATH Keywords: ramified extension
zbMATH Keywords: Brown's constant
author: Sudesh Kaur Khanduja
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.02.028
OpenAlex ID: W2068953007
cites work: ON THE MAIN INVARIANT OF ELEMENTS ALGEBRAIC OVER A HENSELIAN VALUED FIELD
cites work: On Chains Associated with Elements Algebraic over a Henselian Valued Field
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cites work: Roots of Irreducible Polynomials in Tame Henselian Extension Fields
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DOI: 10.1016/J.JPAA.2010.02.028
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description: scientific article; zbMATH DE number 5775718