Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951.
Can we prove the following theorem without Axiom of Choice?
Theorem Let $A$ be a commutative algebra of finite type over a field $k$. Let $I$ be an ideal of $A$. Let $\Omega(A)$ be the set of maximal ideals of $A$. Let $V(I)$ = {$\mathfrak{m} \in \Omega(A)$; $I \subset \mathfrak{m}$}. Let $f$ be any element of $\cap_{\mathfrak{m} \in V(I)} \mathfrak{m}$. Then there exists an integer $n \geq 1$ such that $f^n \in I$.
EDIT So what's the reason for the downvotes?
EDIT What's wrong with trying to prove it without using AC? When you are looking for a computer algorithm for solving a mathematical problem, such a proof may provide a hint. At least, you can be sure that there is a constructive proof.
EDIT To Martin Brandenburg, I think this thread also answers your question.