The algebra and model theory of tame valued fields

TL;DR

This paper develops the algebraic and model-theoretic theory of tame valued fields, establishing foundational principles and completeness results that deepen understanding of their structure and applications.

Contribution

It introduces the algebraic theory of tame fields and proves Ax–Kochen–Ershov Principles, leading to model completeness and insights into valuation theory.

Findings

Proved Ax–Kochen–Ershov Principles for tame fields

Established model completeness relative to value group and residue field

Applied results to Zariski spaces and model theory of large fields

Abstract

A henselian valued field $K$ is called a tame field if its algebraic closure $\tilde{K}$ is a tame extension, that is, the ramification field of the normal extension $\tilde{K}|K$ is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax--Kochen--Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be 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. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.