The model theory of separably tame valued fields

TL;DR

This paper establishes Ax-Kochen-Ershov principles for separably tame valued fields, leading to model completeness and completeness results, and provides new valuation-theoretic proofs for known results in valued field theory.

Contribution

It proves Ax-Kochen-Ershov principles for separably tame fields, extending model theory results and offering new valuation-theoretic proofs for related classes.

Findings

Proved Ax-Kochen-Ershov principles for separably tame fields.

Established model completeness relative to value group and residue field.

Provided alternative proofs for known results in separably closed valued fields.

Abstract

A henselian valued field $K$ is called separably tame if its separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$ is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax-Kochen-Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields.