Subextensions of Separable Field Extension are Separable
Let $E/K/F$ be a tower of fields.
Let $E/F$ be separable.
Then $E/K$ and $K/F$ are separable.
We prove that $E/K$ is separable.
Let $\alpha \in E$.
Let $f$ be its minimal polynomial over $F$.
Let $g$ be its minimal polynomial over $K$.
Then $f$ is separable.
Then $f \in K[x]$ and $f(\alpha) = 0$, hence by definition $g$ divides $f$.
It follows immediately by definition of a separable extension that $K/F$ is.
- Transitivity of Separable Field Extensions, the converse