Composition of Number Fields

Question

If we have number fields $K$ and $L$ containing $\mathbb{Q}$ such that $K\cap L = \mathbb{Q}$, then is it true that $[LK:\mathbb{Q}] = [L:\mathbb{Q}][K:\mathbb{Q}]$?

Answer 1

1. This fact is true if both $K$ and $L$ are Galois extensions of $\mathbb{Q}$, because then the embedding $$ Gal(LK/\mathbb{Q}) \hookrightarrow Gal(K/\mathbb{Q})\times Gal(L/\mathbb{Q}) $$ is an isomorphism.
2. It is not true without the Galois hypothesis : Consider $K = \mathbb{Q}(\sqrt[3]{2})$ and $L= \mathbb{Q}(\omega\sqrt[3]{2})$, then $K\cap L = \mathbb{Q}$, but $[KL:\mathbb{Q}] = 6$, and $[K:\mathbb{Q}][L:\mathbb{Q}] = 3\cdot 3 = 9$.

Answer 2

Certainly not. For example, if $K=L$ and $[K:\mathbb{Q}]>1$, the statement is clearly false. More generally, if $K\subseteq L$, the statement is false. For example, if $K = \mathbb{Q}(\sqrt{2})$ and $L = \mathbb{Q}(2^{1/4})$, then $[LK:\mathbb{Q}] = [L:\mathbb{Q}] = 4$ and $[K:\mathbb{Q}] = 2$.

In general, $[LK:\mathbb{Q}] = [LK:K][K:\mathbb{Q}]$, so that your formula holds if and only if $[LK:K] = [L:\mathbb{Q}]$.