Define the Euler characteristic of a space $X$ to be $$\chi(X)= \sum_i \dim H_i(X, \mathbb Q)$$ This is obviously not necessarily well-defined for an arbitrary space $X$, so let $X$ be a manifold (manifolds have only finitely many nonzero homology groups, and each homology group is finitely generated). I would prefer to keep this question entirely in the realm of closed manifolds.
There's an obvious restatement of this for $\Bbb Q$ replaced by another field $F$, so let $$\chi(M,F) = \sum_i \dim H_i(M, F)$$
Question: When does $\chi(M)=\chi(M,F)$ for all fields $F$? This is true for every finite CW-complex $M$, but it is my impression that not every closed manifold is a finite CW-complex, though I don't have a counter-example. If this is the case (again, for closed manifolds), what is a reference for this fact? If it's false, the question stands. Does the Euler characteristic depend on the field? I'm hoping for either a reference or a counter example.
Edit: The question was resolved below for smooth manifolds via Morse theory, but as far as I can tell this argument is not generally extendable to the topological case (see: Morse theory in TOP and PL categories?). Hopefully there's a known fully topological answer.