Notes on Extremal and Tame Valued Fields

Abstract

Abstract. We extend the characterization of extremal valued fields given in [2] 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 [8] 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 ℵ 1 saturated valued field the valuation is a composition of extremal valuations of rank 1.

Characterizing NIP henselian fields

“…Since this applies also to $$K^{*}v_{p}^{*}$$, it follows that $$(K^{*}v_{p}^{*},\bar{v}^{*})$$ is defectless. By [3, §4], $$(K^{*}v^{*}_{0},\bar{v}_{p}^{*})$$ is maximal (i.e., admits no immediate extensions), so, in particular, is henselian and defectless. Thus, by Lemma 2.9, $$(K^{*}v_{0}^{*},\bar{v}^{*})$$ is defectless.…”

“…We let $$(K^{*},v^{*})$$ be an $$\aleph _{1}$$‐saturated elementary extension of $$(K,v)$$, and consider the standard decomposition for $$(K^{*},v^{*})$$: By Lemma 2.7, $${\Delta }_{0}^{*}/{\Delta }_{p}^{*}$$ is also not discrete. By [3, §4], $${\Delta }_{0}^{*}/{\Delta }_{p}^{*}$$ is isomorphic to $$\mathbb {R}$$. The argument above to show the $$p$$‐divisibility of $${\Delta }_{p}$$ also applies to $${\Delta }_{p}^{*}$$.…”

Decidability in Local and Global Fields

“…The most promising suggestion builds on the notion of extremal valued elds, originally introduced (though with a `wrong' denition) by Yuri Ershov in [Ers04], then, following a suggestion of Sergei Starchenko, the denition was amended and the `correct' denition was put forward in [Ers09] and in [AKP12]. The suggested axiomatization for F q ((t)) given below rst appeared in [AKu16]. Denition 2.3.…”