I found the following problem:
Suppose $K$ is a Galois extension of $F$. Consider $E/F$ a finite extension such that $K\cap E=F$. Show that $[KE:K]=[E:F]$.
Can someone give me a hint? I remember that $[KE:K]\leq[E:F]$ and equality happens when $[K,F],[E,F]$ are relatively prime. But I'm not seeing what forces a Galois extensions to make $[K,F],[E,F]$ relatively prime.