Assume that $F$ is a field and $\operatorname{char}(F)=p$. Let $a$ be an element in $F$ without $p$th root, then the polynomial $x^{p^n}-a$ is irreducible and inseparable over $F$ for all $n$.
I have proved the inseparable part by considering the derivative of the polynomial, but I'm having trouble with the irreducible part. Any help?