Let $k \subset E \subset F$ be a tower of algebraic extensions. If the separable degree $[F:k]_s$ is finite, then $[F:k]_s = [F:E]_s[E:k]_s$.
Here $[F:k]_s$ is defined to be the number of embeddings $F \to \bar k$ fixing $k$.
One essential component of the proof is: If $H: E \to \bar k$ is an embedding of fields fixing $k$, and $G: F \to \bar k$ is an embedding of fields fixing $E$, then they can be assembled to to be a field extension $F \to \bar k$ fixing $k$. How to make sense of this?
Naively I can define an embedding $F \to \bar k$ by sending elements in $E$ according to $H$, and the rest of the elements according to $G$. But this is not necessarily an embedding because $G$ and $H$ might take two elements, one in $E$ and one not in $E$, to the same element in $\bar k$.