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Types in valued fields with coefficient maps. (Types dans les corps valués munis d’applications coefficients.) (French) Zbl 0927.03067

In an earlier paper [L. Bélair and 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.

MSC:

03C60 Model-theoretic algebra
12J10 Valued fields
03C10 Quantifier elimination, model completeness, and related topics
12L12 Model theory of fields

Citations:

Zbl 0857.03018
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