Properties of extensions of algebraically maximal fields
type: Item
instance of: scholarly article
title: Properties of extensions of algebraically maximal fields
zbMATH Open document ID: 1020.03035
published in: Collectanea Mathematica
publication date: 2003-06-30T00:00:00Z
full work available at URL: https://eudml.org/doc/43001
review text: We prove analogues to the Ax-Kochen-Ershov theorem for some cases of pairs of algebraically maximal fields of residue characteristic \(p >0\). By a pair of valued fields, we mean a structure \((K\subset L,v)\), with \((L, v)\) a valued field, and \(K\) a subfield of \(L\). This pair is said to be dense if \(K\) is dense in \(L\) with the topology associated to \(v\). It is said to be separated if, for every finitely generated \(K\)-submodule \(A\) of \(L\), \(v(1-A)\) has a greatest element. Among the consequences of the results are the following two theorems: (1) Let \((K\subset L,v)\) and \((K'\subset L',v')\) be two separated pairs of algebraically maximal fields of positive characteristic (resp. unramiefied fields with perfect residue fields). Assume that the pairs of residue fields are elementarily equivalent and that the pairs of value groups are elementarily equivalent. Then \((K\subset L,v)\) and \((K'\subset L',v')\) are elementarily equivalent. (2) The result in (1) still holds for dense pairs of algebraically maximal Kaplansky fields (resp. finitely ramified fields with perfect residue fields). We generalize those results to some \(n\)-tuples \((K_1 \subset \cdots\subset K_n,v)\) of algebraically maximal fields, and to non-separated non-dense pairs. Then, we deduce from that the decidability of some theories of pairs of algebraically maximal valued fields.
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 12J20
Mathematics Subject Classification ID: 12L05
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 1943363
zbMATH Keywords: pairs of valued fields of positive characteristic
zbMATH Keywords: elementarily equivalent pairs
zbMATH Keywords: decidability
zbMATH Keywords: Ax-Kochen-Ershov theorem
zbMATH Keywords: algebraically maximal fields
author: Gerard Leloup
reviewed by: Gerard Leloup
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1943363