 Solution

In characteristic $p$, the binomial theorem gives $(\alpha +1{)}^p={\alpha }^p+1$. HenceIterating shows that all $\alpha +j$, with $j\in {\mathbb{F}}_p$, are roots. They are distinct, since the prime-field elements $j$ are distinct. There are $p$ of them, equal to the polynomial's degree. Thus if $\alpha \in F$,splits into distinct linear factors in $F[X]$. Equivalently, its derivative is $-1$, so no repeated roots are possible.