Let $f$ be a polynomial of degree $n$ over a field $F$ of characteristic $p$. Suppose $f'=0$. Show that $p\mid n$ and that $f$ has at most $n/p$ distinct roots. I can't solve this question, any help is welcome.
$f'=0$ is equivalent to $f\in F[X^p]$, so $p\mid n$. Furthermore, $f\in F[X^p]$ implies $f(X)=g(X^p)$ where $\deg g=n/p$, so $f$ has at most $n/p$ distinct roots. (Note that $a\neq b$ implies $a^p\neq b^p$ in a field of characteristic $p>0$.)