Proposition 6

— Let L be an extension of a field K. If L is regular, then every subextension of L is regular. Conversely, if every finitely generated subextension of L is regular, then L is regular.

The first assertion follows from Prop. 3, a).

Let M be an extension of K and let $\mathscr{U}$ be the set of all finitely generated subextensions of L. For each E $\in \mathscr{U}$, the ring $M \otimes_K E$ may be identified with a subring of $M \otimes_K L$ and we thus have an increasing directed family of subrings of $M \otimes_K L$ whose union is $M \otimes_K L$. Now the second assertion follows immediately.