Exercise 30, § 5

Let $k$ be a field and $A$ a local, Noetherian and complete $k$-algebra. We say that $A$ is formally smooth if, for every finite extension $k'$ of $k$, the ring $A \otimes_k k'$ is regular. Prove that $A$ is formally smooth if and only if, for every finite radicial extension $k'$ of $k$, the ring $A \otimes_k k'$ is regular (one may use Exer. 14, p. 96 and imitate the proof of Exer. 23, p. 97).