Here you can see part (1). We are now going to prove a more interesting result than the one we proved in part (1). But we need to get prepared first. The following important result is known as Zariski’s lemma.

**Lemma**. (Zariski, 1946) Let be a field and let be a finitely generated commutative algebra. If is a field, then is algebraic over and thus

*Proof*. The proof is by induction over If then and since is a field, Thus for some integer and Then and so and hence is algebraic over Now suppose that If all are algebraic over then is algebraic over and we are done. So we may assume that is transcendental over Since is a field, and thus By the induction hypothesis, is algebraic over So every satisfies some monic polynomial of degree Let be the product of the denominators of the coefficients of and put Let be the maximum of Then multiplying through by shows that each is integral over Note that since is transcendental over the polynomial algebra over Thus I can choose an irreducible polynomial such that

Now because is a field. Thus for a large enough integer we have and hence is integral over But is a UFD and we know that every UFD is integerally closed (in its field of fraction). Therefore which is absurd because then contradicting

**Corollary**. Let be a field and let be a finitely generated commutative algebra.

1) If is a maximal ideal of then

2) If is a field and is algebraically closed, then

3) If is a domain, is algebraically closed and then there exists a ring homomorphism such that for all

*Proof*. 1) Let Then and we are done by the lemma.

2) By the lemma, is algebraic over and thus, since is algebraically closed,

3) Let be the set of all monomials in Clearly is multiplicatively closed and because is a domain. Consider the localization of at Clearly and so is finitely generated. Let be a maximal ideal of Note that for all because each is a unit in Now, we have by 2). Let be a ring isomorphism. We also have the natural ring homomorphism and an inejctive ring homomorphism defined by Let Then and for all because is an isomorphism and for all

See part (3) here.