I am having some trouble proving this statement:
For some field extension $L/M, (L \subset \overline{M})$ we denote by $\Delta_{L/M}$ the M-embeddings of $L$ into $\overline{M}.$
Consider the following extension tower $F \subset K \subset E$, where $K=F(\alpha)$ and $E=F(\alpha,\beta)$ for some $\alpha, \beta \in E$. Also consider $E \subset \overline{F}$, so $\alpha$ and $\beta$ are algebraic over $F$. Then show that
$$ \left| \Delta_{E/F} \right| = \left| \Delta_{E/K} \right|\left| \Delta_{K/F} \right|. $$
I know that $\left| \Delta_{K/F} \right|$ and $\left| \Delta_{E/K} \right|$ will be equal to the number of different roots of the minimal polynomial of $\alpha$ over $F$ and the number of different roots of the minimal polynomial of $\beta$ over $K$, respectively. But I have some difficulties picturing what $\Delta_{E/F}$ looks like.
Any hints ? Thank you.