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## Main Question or Discussion Point

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

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

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