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I'm trying to prove the following:

Suppose $A=\mathbb{Z}[x_1,\ldots,x_n]/I$ where $I$ is some ideal. Then for all $m \in Specm(A)$ we have $\mid A/m \mid$ is finite.

I've seen some proofs of this on the site, but I have the following restriction when proving this. I only know that "If A is a field and is contained in an affine K-domain, then A is algebraic," where K is a field. Are there any proofs of this that do not refer to theorems about Jacobson rings?

user115697
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  • So Zariski lemma is "If K is a finitely generated algebra over a field k and if K is a field, then K is a finite field extension of k." I've proven that "If K is an algebra over field k, K is a field, and K is contained in a finitely generated k-domain, then K is algebraic over k". If K is contained in a finitely generated k-domain, then K must also be finitely generated, right? So I have "If K is a finitely generated algebra over a field k and if K is a field, then K is algebraic over k." But if K is algebraic over k, and finitely generated, then K is a finite field extension of k. – user115697 Feb 24 '14 at 22:26
  • That is, $K=k[a_1,\ldots,a_r]=k(a_1,\ldots, a_n)$ and thus $K:k$ is finite. This gives us Zariski's lemma. Does this look right? If so, how would I go about proving my initial theorem given Zariski lemma? – user115697 Feb 24 '14 at 22:28
  • http://math.stackexchange.com/a/148782/3217 – Georges Elencwajg Feb 24 '14 at 22:53

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