Proposition 4

— Let $A$ be an algebra over field $K$ which is an integral domain, and $E$ its field of fractions. Let $L$ be an extension of $K$; if the ring $L \otimes_K A$ is an integral domain the same is true of $L \otimes_K E$.

If $L \otimes_K A$ is an integral domain, it may be embedded in its field of fractions $F$. Write $u(x) = x \otimes 1$ for $x \in L$ and denote by $v$ the $K$-homomorphism of $E$ into $F$ which extends the injective homomorphism $y \mapsto 1 \otimes y$ of $A$ into $F$. By Prop. 6 (V, p. 14) the subfields $u(L)$ and $v(E)$ of $F$ are linearly disjoint over $K$; therefore the homomorphism $u * v$ of $L \otimes_K E$ into $F$ (V, p. 12) is injective. This shows $L \otimes_K E$ to be an integral domain.