Composition Ax-Kochen/Ershov principles and tame fields of mixed characteristic

TL;DR

This paper investigates conditions under which the theory of a valued field can be determined by its coarsening and residue field theories, establishing positive results for tame equal characteristic fields and counterexamples in mixed characteristic.

Contribution

It extends the understanding of the composition AKE principle, identifying when it holds for tame fields and providing counterexamples in mixed characteristic cases.

Findings

The composition AKE principle holds for tame fields of equal characteristic.

Counterexamples show the principle fails in mixed characteristic.

The theory of a tame mixed characteristic field cannot be solely determined by its components.

Abstract

We study in which settings we have a composition AKE principle, i.e. when the theory of the coarsening $(K,w)$ and the theory of the induced valuation $(Kw,\overline{v})$ determine the theory of the composition $(K,v)$. We show that this is the case when $(K,w)$ is tame of equal characteristic, and provide counterexamples in mixed characteristic. We further show that, for a tame field of mixed characteristic, the theory of the valued field cannot, in general, be determined solely by the theories of its underlying field, its residue field, and its value group.