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### Extremal fields and tame fields

## Abstract

In the year 2003 Yuri Ershov gave a talk at a conference in Teheran on

his notion of ``extremal valued fields''. He proved that algebraically

complete discretely valued fields are extremal. However, the proof

contained a mistake, and it turned out in 2009 through an observation by

Sergej Starchenko that Ershov's original definition leads to all

extremal fields being algebraically closed. In joint work with Salih

Durhan (formerly Azgin) and Florian Pop, we chose a more appropriate

definition and then characterized extremal valued fields in several

important cases.

We call a valued field (K,v) extremal if for all natural numbers n and

all polynomials f in K[X_1,...,X_n], the set of values {vf(a_1,...,a_n)

| a_1,...,a_n in the valuation ring} has a maximum (which is allowed to

be infinity, attained if f has a zero in the valuation ring). This is

such a natural property of valued fields that it is in fact surprising

that it has apparently not been studied much earlier. It is also an

important property because Ershov's original statement is true under the

revised definition, which implies that in particular all Laurent Series

Fields over finite fields are extremal. As it is a deep open problem

whether these fields have a decidable elementary theory and as we are

therefore looking for complete recursive axiomatizations, it is

important to know the elementary properties of them well. That these

fields are extremal could be an important ingredient in the

determination of their structure theory, which in turn is an essential

tool in the proof of model theoretic properties.

The notion of "tame valued field" and their model theoretic properties

play a crucial role in the characterization of extremal fields. A valued

field K with separable-algebraic closure K^sep is tame if it is

henselian and the ramification field of the extension K^sep|K coincides

with the algebraic closure. Open problems in the classification of

extremal fields have recently led to new insights about elementary

equivalence of tame fields in the unequal characteristic case. This led

to a follow-up paper. Major suggestions from the referee were worked out

jointly with Sylvy Anscombe and led to stunning insights about the role

of extremal fields as ``atoms'' from which all aleph_1-saturated valued

fields are pieced together.