When is Galois extension over intersection of subfields finite
Suppose that $K$ and $K^{'}$ are subfields of $L$ such that $ L/K$ (resp. $L/K^{'}$) is a Galois extension with Galois group $G$ (resp. $G^{'}$). Show that $L/(K \cap K^{'})$ is a Galois extension if and only if the group $H$, generated by $G$ and$ G^{'}$, is finite. Show furthermore that if this is so, then $H = Gal(L/(K \cap K^{'}))$.

I would suggest increasing the bounty.

Yes, the bounty is too low for the level of the question.

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