Let $\mathbb{Q} \subset F$ be a field extension. Prove that if $f(x) \in F[x]$ is irreducible, then it has no repeated roots in any field extension of F.
as a hint we were given that a repeated root must be a roof of both $f(x)$ and its derivative. but I'm still not really sure what to do with it. any help is appreciated!
If $f$ has a repeated root $\alpha$, it would be a root of $f'$ too, but $ker$ $ eval_{\alpha}=(f)$ as long as $f$ is irreducible. Then, $f|f'$ which is impossible because $deg(f)>deg(f')$ and $char(\mathbb{Q})=0$.