Let $L$ and $L'$ be fields in $\mathbb{C}$,and $L\lor L'$ be a fields which is smallest one containing $L$, $L'$. I wonder the following is true.
$[L\lor L':L']=[L:L\cap L']$.
Let $\{a_1,...,a_m\}$ and $\{b_1,...,b_n\}$ be basis of $L,L'$. Since $L \lor L' \subset (L \cap L')[a_1b_1,...,a_mb_n]$ and $[(L \cap L')[a_1b_1,...,a_mb_n]:L \cap L']\leq mn$, so I know $[L\lor L':L']\leq[L:L\cap L']$.
For example, if $L$ is a quadratic extension of $L\cap L'$, then $[L\lor L':L']$ is $1$ or $2$. If this is $1$, then $L \lor L' =L'$ so $L \subset L'$, so we have $L=L \cap L'$.
What about any finite extension?
No, it is not true:
If $\omega=e^{2i\pi/3}, L=\mathbb Q(\sqrt [3]2)$ and $L'= \mathbb Q(\omega\sqrt [3]2)$ we have $L\cap L'=\mathbb Q, L\lor L'=\mathbb Q(\sqrt [3]2,\omega )$ so that $$[L\lor L':L']=2\neq [L:L\cap L'] =3$$