0
$\begingroup$

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!

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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$(EF/F)$? $\endgroup$ – El.Gon.Zalo Apr 30 '18 at 11:27
  • $\begingroup$ I searched about this last question and the answer is positive. This is exactly the shift property. $\endgroup$ – El.Gon.Zalo Apr 30 '18 at 11:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.