On the extension of a valuation on a field \(K\) to \(K(X)\). II
type: Item
instance of: scholarly article
title: On the extension of a valuation on a field \(K\) to \(K(X)\). II
zbMATH Open document ID: 0881.12004
author: N. Popescu
author: Constantin Vraciu
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1997-11-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108410
review text: Two types of composite valuations of a simple transcendental extension \(K(X)\) of a valued field \((K,v)\) are characterized. Namely, the residually algebraic extension of the first kind and the composite of two residually transcendental extensions are given in theorems 2.1 and 3.1 respectively. Here a valuation \(u\) of \(K(X)\) is said to be of the first kind if there exists a residual transcendental extension \(w\) of \(v\) to \(K(X)\) such that \(u\leq w\). These results are based on the ``fundamental theorem'' of the characterization of residual transcendental extensions of a valuation. As an application one can describe all the valuations of the field of rational functions in \(n\) independent variables over the valued field \((K,v)\). In this connection see part I [\textit{N. Popescu} and \textit{C. Vraciu}, ibid. 87, 151-168 (1992; Zbl 0774.12009)].
Mathematics Subject Classification ID: 12J20
Mathematics Subject Classification ID: 12F20
zbMATH DE Number: 1021757
zbMATH Keywords: simple transcendental extension of a valued field
zbMATH Keywords: composite valuations
zbMATH Keywords: residual transcendental extension
reviewed by: N. Sankaran
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1021757