Corollary 2
Tag 06W4, permanent. Cite it as [Bourbaki, Tag 06W4].
Algebra, chapter V, § 17. Regular extensions
— Let K be an algebraically closed field. Every K-algebra which is an integral domain is a regular K-algebra. In particular, every extension of K is regular.
This follows from Cor. 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.