Let $k$ be an imperfect field of characteristic $p>0$. Let $a \in k \backslash k^p$.
(1) Show that the polynomial $X^p-a \in k[X]$ is irreducible.
(2) Let $A=k[X] /\left(X^{p^2}-a X^p\right)$. Compute $A_{\text {red }}$, the quotien $\mathrm{t}$ of $A$ by its nilpotent radical.