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 $H = Gal(L/(K \cap K^{'}))$.

Algebra
Jbuck Jbuck
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  • Mathe Mathe

    I would suggest increasing the bounty.

  • Nirenberg Nirenberg

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

  • Jbuck Jbuck

    @Philip Hey there, I hope you're alright! I was wondering if you could help me with some questions this Friday, 10a.m. GMT +1. I will have around a 2-2.5 hour window and I'm willing to pay $60 per question if you're online and able to answer within the time limit, would you be interested? The questions are related to representation theory.

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Mathe Mathe
The answer is accepted.