The first paragraph of above excerpt from Lang's book I have understood easily.
But I cannot understand one moment from the second one: if $f$ is separable polynomial in $k[X]$ with $f(\alpha)=0$, the we can consider $f$ as polynomial from $F[X]$. But in order to conlude that $\alpha$ is separable over $F$ (here we are using the first paragraph) we need to show that $f$ is separable over $F$.
If $k\subset F$ and $f$ is separable over $k$ how to show that $f$ is separable over $F$?
I have tried some approaches but did not get smth good.
Would be grateful for help!