Tame fields and tame extensions
type: Item
instance of: scholarly article
title: Tame fields and tame extensions
zbMATH Open document ID: 0910.12003
published in: Journal of Algebra
publication date: 1998-10-21T00:00:00Z
review text: Let \(V\) be a henselian valuation of any rank of a field \(K\), and let \(\overline V\) be the extension of \(V\) to a fixed algebraic closure \(\overline K\) of \(K\). In this paper, the author proves that \((K,V)\) is a tame field, i.e., every finite extension of \((K,V)\) is tamely ramified, if and only if \(\forall\alpha\in\overline K\setminus K\), there exists \(a\in K\) such that \(\overline V(\alpha-a)\geq\Delta_K(\alpha)\), where \(\Delta_K (\alpha)= \min\{ \overline V(\alpha'-\alpha)|\alpha'\) varies over all \(K\)-conjugates of \(\alpha\}\). In the course of proving the main result, the author shows that if \((K',V')\) is any finite Galois tame extension of a henselian valued field \((K,V)\), then to each \(\alpha\in K'\) there corresponds \(a\in K\) such that \(V'( \alpha-a)\geq\Delta_K(\alpha)\). The author gives an example showing that if \((K', V')\) is a finite Galois extension of a henselian field \((K,V)\), then it is not necessarily the case that the statement to each \(\alpha\in K'\) there corresponds \(a\in K\) such that \(V'(\alpha-a)\geq\Delta_K(\alpha)\) implies the statement \((K', V')\) is a tame extension of \((K,V)\). However the author shows that the implication holds under the added assumption that \(\Delta_K(\alpha)= \omega_K (\alpha)\) for some generator \(\alpha\) of \(K'/K\), where \(\omega_K(\alpha)=\infty\) if \(\alpha\in K\) and \(\omega_K(\alpha)=\max\{\overline V(\alpha-\alpha')\mid \alpha' \neq \alpha\) varies over all \(K\)-conjugates of \(\alpha\}\) otherwise.
reviewed by: John N. Mordeson
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 1214118
zbMATH Keywords: henselian valuation
zbMATH Keywords: tame field
zbMATH Keywords: finite Galois tame extension
author: Sudesh Kaur Khanduja
MaRDI profile type: Publication
full work available at URL: jabr.1997.7298
OpenAlex ID: W2042084659
cites work: Zeros of polynomials over local fields. The Galois action
cites work: Q5521595
cites work: Valuation theory
cites work: A note on a result of J. Ax
cites work: On a result of James Ax
cites work: Simple transcendental extensions of valued fields. II: A fundamental inequality
DOI: 10.1006/JABR.1997.7298
description: scientific article; zbMATH DE number 1214118