Types in valued fields with coefficient maps
type: Item
instance of: scholarly article
title: Types in valued fields with coefficient maps
zbMATH Open document ID: 0927.03067
author: Luc Bélair
published in: Illinois Journal of Mathematics
publication date: 1999-11-23T00:00:00Z
review text: In an earlier paper [\textit{L. Bélair} and \textit{M. Bousquet}, C. R. Acad. Sci., Paris, Sér. I 323, 841-844 (1996; Zbl 0857.03018)], Delon's analysis of types in valued fields is transposed in a formalism which includes a coefficient map, i.e. a homomorphism from the multiplicative group of the base field into the multiplicative group of the residue field which extends the residue map on the units of the valuation ring. Things become transparent, especially coheirs and the independence property. In this paper, we use a generalization of coefficient maps due to van den Dries to study the case of unramified henselian valued fields of characteristic zero with a residue field of positive characteristic \(p\). In particular we still obtain: the base valued field has the independence property iff the residue field has it. Let \(I\) be an ideal of the valuation ring \(V\). A coefficient map of order \(I\) is a homomorphism from the multiplicative group of the base field into the multiplicative group of the quotient ring \(V/I\) which extends the canonical quotient map on the units of the valuation ring. We use the family of ideals \(p^n V\). We prove a quantifier elimination result for this formalism, relative to the base field elements.
reviewed by: Luc Bélair
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 1333073
zbMATH Keywords: types in valued fields
zbMATH Keywords: coefficient map
zbMATH Keywords: unramified henselian valued fields
zbMATH Keywords: residue field
zbMATH Keywords: independence property
zbMATH Keywords: valuation ring
zbMATH Keywords: quantifier elimination
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1333073