Superrosy fields and valuations

TL;DR

This paper proves that non-trivial valuations on infinite superrosy fields of positive characteristic have divisible value groups and algebraically closed residue fields, extending to a broader class of fields with specific algebraic properties.

Contribution

It establishes a general result linking algebraic properties of fields to valuation theory, particularly for superrosy fields and fields with finite extension index conditions.

Findings

Non-trivial valuations on superrosy fields have divisible value groups.

Residue fields of such valuations are algebraically closed in positive characteristic.

Fields with certain algebraic finiteness conditions either admit a definable valuation or have valuations with specific properties.

Abstract

We prove that every non-trivial valuation on an infinite superrosy field of positive characteristic has divisible value group and algebraically closed residue field. In fact, we prove the following more general result. Let $K$ be a field such that for every finite extension $L$ of $K$ and for every natural number $n>0$ the index $[L^*:(L^*)^n]$ is finite and, if $char(K)=p>0$ and $f: L \to L$ is given by $f(x)=x^p-x$, the index $[L^+:f[L]]$ is also finite. Then either there is a non-trivial definable valuation on $K$, or every non-trivial valuation on $K$ has divisible value group and, if $char(K)>0$, it has algebraically closed residue field. In the zero characteristic case, we get some partial results of this kind. We also notice that minimal fields have the property that every non-trivial valuation has divisible value group and algebraically closed residue field.