Corollary 1

Let $k$ be a field. The tensor product of two absolutely regular (resp. absolutely normal) $k$-algebras, one of which is essentially of finite type, is an absolutely regular (resp. absolutely normal) $k$-algebra.

Let $A$ and $B$ be two $k$-algebras satisfying the hypotheses of the corollary. Let $k'$ be a finite radicial extension of $k$. The ring $B_{(k')}$ is regular (resp. normal); the ring $A \otimes_k B_{(k')}$ is regular (resp. normal) by Prop. 7, as is the ring $(A \otimes_k B) \otimes_k k'$ which is isomorphic to it.