# How to show an affine variety in $\mathbb{A}^{n}$ either has finite cardinality or equal to that of $k$?

How to show an affine variety in $\mathbb{A}^{n}$ either has finite cardinality or equal to that of $k$? I do not know how to solve this via elementary methods.

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are you allowed to use Noether normalization? –  user29743 Nov 12 '12 at 0:52
I am not working on homework, but Noether normalization only applies to projective varieties right? –  Bombyx mori Nov 12 '12 at 0:53
I take a look by googling, and found the popular version is different from what I know. Still, I do not know how to prove it. –  Bombyx mori Nov 12 '12 at 0:55
I'm guessing $k$ is a field, but when you write $\mathbb{A}^k$, it looks like it could also be an integer. Is this notation supposed to mean the affine line over $k$? I think it might be a bit more clear if you used the notation $\mathbb{A}_k^1$ (if this is what you mean). –  Keenan Kidwell Nov 12 '12 at 1:20
Sorry for the confusion, clarified; also by Noether normalization lemma this is immediate. –  Bombyx mori Nov 12 '12 at 1:21