Exercise 3, Appendix 0

Let $k$ be an imperfect field of characteristic $p > 0$ and $R = k[T]_{(T)}$. Let $a$ be an element of $k$ which is not a $p$-th power. Now prove that the ring $A = R[X]/(X^p - aT^p)$ is local, integral, and has residue field $k$. Now prove that the ring $B = A[Y]/(Y^p - a)$, which is an inflation of $A$, is not reduced.

APP.

EXERCISES

AC IX.85