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Relative elimination of quantifiers for Henselian valued fields. (English) Zbl 0734.03021
Let L and F be two Henselian unramified fields of characteristic zero, e.g. p-adic fields \({\mathbb{Q}}_ p\), formal power series fields F((t)) with F of characteristic zero etc. In the 1960s Ax-Kochen and Ershov proved that L and F are elementarily equivalent (that is, they satisfy the same sentences in the usual language of valued fields) if and only if both their value fields and residue groups are (elementarily equivalent).
This led to a powerful elimination theory for such fields which has given many applications in Number Theory and Algebra. The most general type of elimination is the so-called relative elimination of quantifiers which is the reduction of statements about such a field to an equivalent statement which is a Boolean combination of statements of a prespecified simple type; what is meant here by “simple” can be one of several things, and each different interpretation denotes a different approach to the subject, often with different applications. For example, one may consider simply the statements about the value group and the residue field.
The main result of the paper is to give a necessary and sufficient condition for two such fields L and F that extend a common subfield K to be elementarily equivalent; it is then proved that a number of results about existence of relative elimination of quantifiers follow.
Particular features of the paper in contrast to earlier approaches are the uniformity of its methods as well as its strong algebraic flavor (only basic notions from model theory are used).

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