Sous-corps fermés d'un corps valué
type: Item
instance of: scholarly article
title: Sous-corps fermés d'un corps valué
zbMATH Open document ID: 0544.12016
DOI: 10.1016/0021-8693(84)90186-8
author: Michel Matignon
author: Marc Reversat
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: K is a complete non-Archimedean valued field and K(X) denotes a transcendental extension of K. The valuations on K(X), extending the valuation of K are classified. For a fixed valuation on K(X), one makes a valuation on the algebraic closure \(\overline{K(X)}\) of K(X) and \(\overline{K(X)}{\hat{\;}}\) is the completion of \(\overline{K(X)}.\) The main results of the paper are: Theorem 1. If the residue field of K has characteristic 0 then the only algebraically closed and complete fields between K and \(\overline{K(X)}{\hat{\;}}\) are \=K{\^{\ }} and \(\overline{K(X)}{\hat{\;}}\). - Theorem 2. If the residue field of K has positive characteristic then there are infinitely many algebraically closed and complete fields between K and \(\overline{K(X)}{\hat{\;}}.\) A special case of Theorem 2 is: \(\overline{{\mathbb{F}}_ p((T))}{\hat{\;}}\) contains infinitely many algebraically closed and complete subfields. The two theorems have implications for the continuous Galois group of \(\overline{K(X)}{\hat{\;}}\) over K. In the proofs an important rôle is played by the different of an infinite field extension.
Mathematics Subject Classification ID: 12J25
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12F15
zbMATH DE Number: 3865436
zbMATH Keywords: extensions of valuations
zbMATH Keywords: complete non-Archimedean valued field
zbMATH Keywords: transcendental extension
zbMATH Keywords: continuous Galois group
zbMATH Keywords: different of an infinite field extension
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90186-8
OpenAlex ID: W1985954180
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description: scientific article; zbMATH DE number 3865436