Let $F \subseteq E$ be an extension field.
A polynomial $f\in F[x]$ is separable if every irreducible factor of $f$ has distinct roots. An element $\alpha \in E$ is separable if the minimal polynomial of $\alpha$ over $F$ is separable. An extension field $E$ is separable if every element in $E$ is separable.
My question is as follows: Let $\alpha \in E$ be a separable element. I want to prove or disprove that $F[\alpha]$ is a separable extension.