Corollary

Let k be a field and R and S two integrally closed k-algebras. Suppose that the ring $R \otimes_k S$ is an integral domain and that the fields of fractions K and L of R and S respectively are separable over k. Then the ring $R \otimes_k S$ is an integrally closed domain.

As R and S are identified with subalgebras of $R \otimes_k S$, K and L are identified with subfields of the field of fractions $\Omega$ of $R \otimes_k S$ which are linearly disjoint over $k$ (Algebra, Chapter V, § 2, no. 3, Proposition 5). It then follows from Proposition 19 that $R \otimes_k L$ and $K \otimes_k S$ are integrally closed domains; as their intersection is $R \otimes_k S$ (Chapter I, § 2, no. 6, Proposition 7), $R \otimes_k S$ is an integrally closed domain (no. 2, Corollary to Proposition 8).

\* Given two irreducible affine varieties V, W defined over $k$, their product $V \times W$ is an affine variety and the ring of functions regular on $V \times W$ is identified with the tensor product over $k$ of the ring of functions regular on V and the ring of functions regular on W. The Corollary to Proposition 19 shows that, if V and W are normal over $k$, then $V \times W$ is normal over $k$. \*