Type-definable NIP fields are Artin-Schreier closed

TL;DR

This paper proves that infinite fields with NIP in characteristic p are closed under Artin-Schreier extensions, extending previous results and impacting the understanding of field extensions in model theory.

Contribution

It generalizes a theorem by Kaplan, Scanlon, and Wagner, showing that type-definable NIP fields of characteristic p are Artin-Schreier closed.

Findings

NIP fields of characteristic p are Artin-Schreier closed

No Artin-Schreier extensions exist for such fields

p does not divide degrees of finite separable extensions

Abstract

Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p > 0$, then $K$ is Artin-Schreier closed (it has no Artin-Schreier extensions). As a consequence, $p$ does not divide the degree of any finite separable extension of $K$. This generalizes a theorem of Kaplan, Scanlon, and Wagner.