Let $K$ and $L$ be extensions of $F$. Show that $KL$ is Galois over $F$ if both $K$ and $L$ are Galois over $F$. Is the converse true?
I should show $|Gal(KL/F)|=[KL:F]$ and for this I am using $|Gal(K/F)|=[K:F], |Gal(L/F)|=[L:F] $ and $[KL:F]\leq[K:F][L:F]$, and I get to the next but I do not know what else to do, could someone help me please? Is there a counterexample to the converse? Thank you in advance.