Let $F$ be a finite field. Then any $f \in F[x]$ is separable.
How do you tackle this?
I thought of letting $f \in F[x]$ be irreducible, and let $\alpha$ be a root of $f$. Towards a contradiction, assume that $\alpha$ is a repeated root. Then $\alpha$ is also a root of the formal derivative of $f$ ; $D(f)$.
Since $f$ is irreducible, it must be the minimal polynomial of $\alpha$ over $F$. But $\alpha$ is a root of $D(f)$ which has degree less than $f$. I think this is a contradiction, but I’m not sure?