Valued fields with contractive automorphism and Kaplansky fields
type: Item
instance of: scholarly article
title: Valued fields with contractive automorphism and Kaplansky fields
zbMATH Open document ID: 1218.12004
author: Salih Azgin
published in: Journal of Algebra
publication date: 2011-01-21T00:00:00Z
review text: The author determines the elementary theories of valued fields equipped with a contractive automorphism. This is achieved under certain technical assumption on the residue difference field. Precisely, he proves the following: Theorem: Let \(K\) and \(K'\) be \(\sigma\)-Henselian valued difference fields with contractive distinguished automorphisms. Suppose \(K\) and \(K'\) have residue difference fields \(k\) and \(k'\), of characteristic \(0\), and value difference groups \(\Gamma\) and \(\Gamma'\) respectively. Then \(K\equiv K'\) if and only if \(k\equiv k'\), as difference fields, and \(\Gamma\equiv\Gamma'\), as ordered difference groups. The techniques involved can also be applied to valued fields of positive characteristic to obtain alternative proofs of well-known results about Kaplansky fields. For example, the following: Theorem: If \(K\) is a Kaplansky field, then \(K\) has, up to isomorphism over \(K\), a unique algebraically maximal, immediate, algebraic extension.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 5840419
zbMATH Keywords: valuation theory
zbMATH Keywords: model theory
zbMATH Keywords: valued difference fields
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.08.003
OpenAlex ID: W2048821031
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DOI: 10.1016/J.JALGEBRA.2010.08.003
description: scientific article; zbMATH DE number 5840419