Let $L, K, F = L \cap K$ be fields such that $[L:F] , [K:F] < \infty$. $LK$ is defined as the compositum of the two fields(shortest field containing $K$ and $L$ in some algebraic closure).
Also, assume hat $L$ is Galois. Then, I can prove that $[L:F][K:F] = [LK:F]$ assuming that $L = F[\theta]$ using something like the primitive element theorem.
Does the result hold in more generality($L \neq F[\theta])$ and how does one prove it?