If F is a field of characteristic 0 with subfields K, L such that F is the compositum of K and L and [ L : L ∩ K ] is prime, must be K and L be linearly disjoint over L ∩ K?
In other words, must [ KL : K ] = [ L : L ∩ K ] if [ L : L ∩ K ] is prime?
Here K, L are fields of characteristic 0 (definitely not p), but I don't assume [ K : K ∩ L ] is finite (or normal or anything). I think it doesn't have to be true if that index [ L : L ∩ K ] is not prime and the field extension K ∩ L ≤ K is not normal. I am hoping it is true when the index [ L : L ∩ K ] is prime.
The motivation is to show either every non-identity p-subgroup of PGL(2, K) is reducible or every non-identity p-subgroup of PGL(2, K) is irreducible.