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
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    I would suggest increasing the bounty.

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

  • Jbuck Jbuck
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Mathe Mathe
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The answer is accepted.
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