Notes on extremal and tame valued fields

TL;DR

This paper extends the characterization of extremal valued fields to mixed characteristic cases with perfect residue fields, providing a complete classification of tame extremal fields and exploring their properties and constructions.

Contribution

It offers a complete characterization of tame extremal valued fields in mixed characteristic and introduces model theoretic results and examples related to extremal fields.

Findings

Complete characterization of tame extremal valued fields.

Images of additive polynomials have the optimal approximation property in extremal fields.

Construction methods for extremal valued fields, including in saturated fields.

Abstract

We extend the characterization of extremal valued fields given in \cite{[AKP]} to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finite $p$-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in \cite{[K1]} for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every $\aleph_1$ saturated valued field the valuation is a composition of extremal valuations of rank 1.