Exercise 1, § 6

Let $k$ be a non-perfect field of characteristic $p > 2$, $a$ an element of $k - k^p$, A the ring $k[X, Y]/(X^2 - Y^p + a)$.

a) Prove that the ring A is regular (one may prove that A is integrally closed in its field of fractions $K = k(Y)[X]/(X^2 - Y^p + a)$).

b) Prove that $k$ is algebraically closed in K.

c) Let $k'$ be the extension $k(a^{1/p})$ of $k$; prove that the ring $A \otimes_k k'$ is not normal.