12.69 (1992)

Let $F$ be a countably infinite field and $G$ a finite group of automorphisms of $F$. The group ring $\mathbb{Z}G$ acts on $F$ in a natural way. Suppose that $S$ is a subfield of $F$ satisfying the following property: for any $x \in F$ there is a non-zero element $f \in \mathbb{Z}G$ such that $x^f \in S$. Is it true that either $F = S$ or the field extension $F/S$ is purely inseparable? One can show that for an uncountable field an analogous question has an affirmative answer.