Corollary 3

Tag 04RO, permanent. Cite it as [Bourbaki, Tag 04RO].

Commutative Algebra, chapter V, § 3. Finitely generated algebras over a field

Let $A$ be a Jacobson ring. If there exists a finitely generated $A$ -algebra $B$ containing $A$ and which is a field, then $A$ is a field and $B$ is an algebraic extension of $A$ .

It suffices to apply Theorem 3 (ii) with $m' = (0)$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Theorem 3