Conjugate fields and conjugate subgroups of an automorphism group

imurme8
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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?
 
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what is the definition of conjugate fields? if you give it precisely maybe you can answer your own question.
 

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