Suppose I have an algebraic field extension $L/F$. What do I call the fixed field of $\operatorname{Aut}(L/F)$? It's sort of like the mirror image of the Galois closure: the smallest field $K>F$ such that $L/K$ is Galois.
As a particular example, what is the fixed field of $\operatorname{Aut}(\overline{F_p(t)}/F_p(t))$?