Let $F $ be a field with char $F =p >0$. If $\alpha \in K/F$ is separable over $F $, prove that $F (\alpha)/F $ is separable.

Question

Let $F $ be a field with char $F =p >0$. If $\alpha \in K/F$ is separable over $F $, prove that $F (\alpha)/F $ is separable.

I am really unsure how to solve this. I know that $K/F (\alpha) $ is separable (please correct me if I am wrong) but how can I show the other direction?