Maximal separable intermediate fields of large codegree

Abstract:

Let $k$ be a function field in $n(n > 0)$ variables over ${k_0}$ a field having characteristic $p \ne 0$. An intermediate field $s$ is maximal separable if $s/{k_0}$ is separable and $s$ is not properly contained in any subfield of $k$ separable over ${k_0}$. The following result is proved. If $n = 1$ the set $\Delta = \{ [k:s]|s$ maximal separable} is bounded if and only if the algebraic closure ${\bar k_0}$ of ${k_0}$ in $k$ is separable over ${k_0}$. If $n \geqslant 1$ and $\Delta$ is bounded then ${\bar k_0}/{k_0}$ is separable. An upper bound for $\Delta$ is obtained for the case $n = 1$ and ${\bar k_0}/{k_0}$ separable.