Let $F<E$ be an algebraic field extension. Let $\alpha\in E$ be such that $\min(\alpha,F)$ has only one root in $E$ (which will be $\alpha$).Is it true that for any $p(x)\in F[x]$ we must have:

"$\min(p(\alpha),F)$ has only one root in $E$"

Another question: Does the above conjecture at least hold in characteristic $0$ ?

Thank you a lot.

(Note: $\min(\alpha,F)$ denotes the minimal monic polynomial of $\alpha$ over $F$)