Exercise 4, § 13

Let K be a commutative field of characteristic $p >0$, and let $a$ be an element of K - $K^p$. Denote the (finite-dimensional) K-algebra $K[X]/((X^p-a)^2)$ by A. Prove that the radical $\mathfrak{r}$ of A has square zero, that the K-algebra $A/\mathfrak{r}$ is isomorphic to the radical extension $L = K[X]/(X^p-a)$ of K, and that A does not contain any K-subalgebras isomorphic to L.