weak Nullstellensatz

Statement

Let be an algebraically closed field. Then every maximal ideal (ring theory) of the polynomial ring is of the form

Proof

Let be a maximal ideal and denote as for the remainder of the proof. We can construct the residue field which is finitely generated as a -algebra (since by repetition of Hilbert Basis Theorem the is Noetherian ring so is finitely generated, thus the quotient is finitely generated).

Therefore, by Zariski’s Lemma, is finite, and thus algebraic.

is algebraically closed and is algebraic, so .

Let be the image of under the homomorphism

Therefore , and since is maximal, they are equal.

Corollary

Let be a collection of polynomial functions on with no common zeros. Then the .

Proof

Assume by contradiction we have . Then by Zorn’s lemma for some maximal ideal .

By the theorem above, . Therefore, for all . This is common zero and thus a contradiction.