Type-definable NIP fields are Artin–Schreier closed
Volume 260 / 2023
Fundamenta Mathematicae 260 (2023), 251-261
MSC: Primary 03C60; Secondary 03C45.
DOI: 10.4064/fm149-8-2022
Published online: 14 November 2022
Abstract
Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p \gt 0$, 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.
