Let $f(x)$ be an irreducible polynomial in F which the leading coefficient is 1 and its degree is larger than 2. Show that if $f(x)$ has the same root in its splitting field, then the characteristic of F is $p>0$(prime), and $f(x)=x^{p^n}-a \in F[x], n\ge 1$.
I'm confused how to solve this question using the separable extension, please give me some hints. Thanks!