I want to show "If $E_1$ and $E_2$ are normal extensions of $F$, then the compositum $E_1 E_2$ and $E_1\cap E_2$ are normal extensions."
I already prove this, but I find another proof using an embedding $\sigma$ on $\overline{F}$(not F-embedding).
It says $\sigma (E_1 E_2)=\sigma (E_1) \sigma (E_2)$. I think it holds because $\sigma$ is an embedding, but it is hard to prove it. Do I need to use normalities of $E_i$?
Moreover, more generally, for embedding $\sigma$ on $\overline{F}$, $ \sigma(F(S))=\sigma(F)(\sigma(S))$?
(I know it is holds for $\sigma(E_1\cap E_2)$..)