Type-definable NIP fields are Artin–Schreier closed

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.