Types in valued fields with coefficient maps
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.
scientific article; zbMATH DE number 1333073