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I have a question on Galois theory.

Every field here introduced is a (not necessarily finite) extension of $\mathbb{Q}$. Let $L/K$ be a Galois extension. Let $E,F$ be two intermediate fields, that is $L/E/K$ and $L/F/K$. Is it true that the extension $E/(E\cap F)$ is Galois? If so, is it true that we have Gal$(E/(E\cap F)) \cong $ Gal$(L/F)$??

My attempt: I was trying to use the shift property of Galois extensions, but its statement is actually backwards. Maybe it is necessary to have that $L=EF$ is the compositum of the two subfields?

Thanks in advance!

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No I believe, it should not be Galois. Boring example but take, $K=F=\mathbb{Q}$ and $E=\mathbb{Q}(\sqrt[3]{2})$ and $L=\mathbb{Q}(\sqrt[3]{2},\mu_3)$. $L/K$ Galois and $E$, $F$ are both intermediate fields but $E/(E\cap F)=\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ which is clearly not Galois.

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  • $\begingroup$ Assuming then that $E/(E\cap F)$ is Galois and that $L=EF$, do you think that Gal$(E/(E\cap F))$ is isomorphic to Gal$(EF/F)$? $\endgroup$ Apr 30, 2018 at 11:27
  • $\begingroup$ I searched about this last question and the answer is positive. This is exactly the shift property. $\endgroup$ Apr 30, 2018 at 11:52

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