Completions of r.a.t.-valued fields of rational functions
type: Item
instance of: scholarly article
title: Completions of r.a.t.-valued fields of rational functions
zbMATH Open document ID: 0813.12006
author: Alexandru Zaharescu
author: Adrian Iovita
published in: Journal of Number Theory
publication date: 1995-03-26T00:00:00Z
review text: Let \(\mathbb{Q}_ p\) denote the field of \(p\)-adic numbers, \(\Omega\) its algebraic closure, and \(\overline{\Omega}\) the completion of \(\Omega\) with respect to the unique extension of the \(p\)-adic valuation \(v\) at \(\Omega\). This paper deals with the structure of complete fields \(L\), \(\mathbb{Q}_ p \subseteq L \subseteq \overline{\Omega}\). Let \(F_ c(\overline{\Omega}/\mathbb{Q}_ p)\) denote the set of all complete intermediate fields of \(\overline{\Omega}/\mathbb{Q}_ p\). If \([L:\mathbb{Q}_ p]\) is finite, there exists \(a \in L\) such that \(L = \mathbb{Q}_ p(a)\). The authors show in this paper that if \([L : \mathbb{Q}_ p]\) is infinite, then there exists \(z \in L\) (\(z\) transcendental over \(\mathbb{Q}_ p\)) such that \(L\) is the completion of \(\mathbb{Q}_ p(z)\) with respect to the restriction of the valuation \(v\) of \(\overline{\Omega}\) to \(\mathbb{Q}_ p(z)\). They use the result that the correspondences \(F(\Omega/K) \supset l \to \overline{l} \in F_ c(\overline{\Omega}/K)\) and \(F_ c(\overline{\Omega}/K) \supset L \to L \cap \Omega \in F(\Omega/K)\) are one-to-one and inverse to each other. The authors also prove that \(L\) is the completion of \(Q(x)\) with respect to a valuation \(w\), which extends \(\nu\) to \(Q(x)\) where \(x\) is transcendental over \(Q\). The valuation \(w\) is the residual algebraic torsion extension of \(v\).
reviewed by: John N. Mordeson
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12J20
zbMATH DE Number: 729071
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: residual algebraic torsion extension
zbMATH Keywords: completion
zbMATH Keywords: \(p\)- adic valuation
zbMATH Keywords: complete fields
MaRDI profile type: Publication
full work available at URL: jnth.1995.1014
OpenAlex ID: W1982943651
DOI: 10.1006/JNTH.1995.1014
description: scientific article; zbMATH DE number 729071