Field automorphisms associated with a Galois group (reviewed)

If $G$ is a transitive subgroup of $S_n$, then it can arise as the Galois group of a field extension $L/K$. Let $G_1=\{\sigma\in G\mid \sigma(1)=1\}$, and let $F$ be the fixed field of $G_1$; then $F/K$ is a field extension of degree $n$ whose Galois closure is $L/K$.

Although $F$ might not be Galois over $K$, one can consider the group $\Aut(F/K)$. Via the Galois correspondence, this can be computed group-theoretically; its order equals the order of the centralizer of $G$ in $S_n$.