Let $f(x)$ be an irreducible polynomial of degree $n$ defined over a field $F$, and let $G$ be the Galois group of $f$, identified as a transitive subgroup of $S_n$. Let $K/F$ be the stem field of $f$. We show the automorphism group of $K/F$ is isomorphic to the centralizer of $G$ in $S_n$. We include two applications to computing Galois groups; one in the case $F$ is the rational numbers, the other when $F$ is the 5-adic numbers.
"Centralizers of Transitive Permutation Groups and Applications to Galois Theory." Missouri J. Math. Sci. 27 (1) 16 - 32, November 2015. https://doi.org/10.35834/mjms/1449161364