Corollary 3

Let k be a field and A a noetherian local k-algebra whose residue field is equal to k. In order that A be regular, it is necessary and sufficient that its completion $\hat{A}$ be isomorphic to a formal power series k-algebra $k[[X_1, ..., X_n]]$.

This follows from the equivalence of (i) and (iv) in Theorem 1, and from Proposition 11 of III, § 2, No. 9.