Suppose $char(K)=p$. Let $a\in K$. If $a$ has no $p-$th root in $K$, then show that $X^{p^n}-a$ is irreducible in $K[x]$ for all positive integers $n$.
What I assumed is that $b$ is a $p^n$th root of $a$ in some algebraic closure of $K$. Then we have $X^{p^n}-a=(X-b)^{p^n}$. Now if $X^{p^n}-a$ is reducible, then assume $(X-b)^{k} \in K[X]$ but after that I am not sure how to conclude. Should I consider the coefficients of $X^{k-1}$ or that of the constant term? Which would help?