Maximal separable subfields of bounded codegree

Abstract:

Let $L$ be a function field in $n > 0$ variables over a field $K$ of characteristic $p \ne 0$. An intermediate field $S$ is maximal separable if $S$ is separable over $K$ and every subfield of $L$ which properly contains $S$ is inseparable over $K$. This paper examines when $[L:S]|S$ is maximal separable is bounded. The main result states that this set is bounded if and only if there is an integer $c$ such that any intermediate field ${L_1}$ over which $L$ is purely inseparable and $[L:{L_1}] > {p^c}$ must be separable over $K$. Examples are also given where the above bound is ${p^{n + 1}}$ for any $n \geqslant 1$.