A criterion for p-henselianity in characteristic p

TL;DR

This paper establishes a criterion for p-henselianity in valued fields of characteristic p, showing it is equivalent to every positive valuation element being expressible as x^p - x for some x in the field.

Contribution

The paper provides a new characterization of p-henselianity in characteristic p valued fields based on elements of the form x^p - x.

Findings

p-henselianity is characterized by elements of the form x^p - x

The criterion applies specifically to valued fields of characteristic p

The result offers a practical test for p-henselianity in such fields

Abstract

Let $p$ be a prime. In this paper we give a proof of the followingresult: A valued field $(K,v)$ of characteristic $p \textgreater{} 0$ is$p$-henselian if and only if every element of strictly positivevaluation if of the form $x^p - x$ for some $x \in K$.