Proposition 5

— Every pure extension of a field $K$ is regular.

By the preceding corollary it is enough to prove that every polynomial algebra $A = K[X_i]_i$, is a regular $K$-algebra. Let $L$ be an extension of $K$; the ring $L \otimes_K A$ is isomorphic to $L[X_i]_{i \in I}$ (III, p. 449, Remark 2), and hence an integral domain (IV, p. 9, Prop. 8).