Steve Kleiman  18.704 Homework 1  September 7, 2006

p.21: 10. Let ℤ^{n} ⊂ ℂ^{n} consist of those points with integer coordinates. Prove that ℤ^{n} is not an affine variety. Hint: See Exercise 6 from §1.

Solution: Suppose to the contrary that ℤ^{n} = V(f_{1}, …, f_{s}). Then each f_{i} vanishes at every point of ℤ^{n}, and so f_{i} is the zero polynomial by Exercise 6a from §1. Hence V(f_{1}, …, f_{s}) = ℂ^{n}, a contradiction!