Exercise 22, § 5

a) Let $k$ be a field, $A$ a $k$-algebra of type fini, $k'$ a finite and separable extension of $k$. Prove that if $A$ is regular, then $A \otimes_k k'$ is regular. (One may use Exercise 14, b), p. 96.) b) With the same hypotheses on $A$, let $k'$ be a finite and radicielle extension of $k$. Prove that in general $A \otimes_k k'$ is not regular (take $A = k'$).