I'm working through Atiyah and Macdonald, and I'm stuck on Chapter 1 Problem 27. It asks us to prove that for algebraically closed $k$, we have that all maximal ideals of the ring of polynomials in $n$ variables over $k$ are expressible as $\mathfrak{m}_x$, the set of polynomials vanishing at $x$, for some $x\in k^n$.
In order to prove this via the Internet, I've been able to show a couple of things: first, I've shown that the quotient field of the polynomial ring over the maximal ideal must be a finitely-generated $k$-algebra, that if such an ideal exists, then there is some $i$ such that our ideal doesn't contain $x_i-a_i$ for any $i$.
At this point, I'm not sure where to proceed. A lot of proofs seem to require that $k$ is uncountable, but this doesn't account for all cases. Where should I go from here?