# Conjugate fields and conjugate subgroups of an automorphism group

## Main Question or Discussion Point

Suppose $E$ and $D$ are both finite extensions of $F$, with $K$ being the Galois closure of $\langle D,E \rangle$ (is this the correct way to say it?) Is it correct that $E$ and $D$ are conjugate fields over $F$ iff $G,H$ are conjugate subgroups, where $G,H\leqslant \text{Aut}(K/F)$ are the subgroups which fix $E,D$?

I want to claim also that given $E,D$, we have that their Galois closure $K$ is exactly the field fixed by the core of $\text{Aut}(D/F)$ and $\text{Aut}(E/F)$, but I'm not sure if the "core" is well-defined in this case, since we've not defined a group of which $\text{Aut}(D/F)$ and $\text{Aut}(E/F)$ are a subgroup. What do you think?