Does the Euler characteristic of a manifold depend upon the field of coefficients?

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.

Every topological compact manifold can be embedded into $\mathbb{R}^{n}$ and since it is locally contractible it follows that it is an Euclidean Neighbourhood Retract. The proofs are elementary.

It follows that it is a retract of a finite simplicial complex and so its integral homology groups are finitely generated. This is enough to ensure that the Euler characteristic is independant from the field of coefficients (for example by universal coefficient theorem).

For references, see Appendix A of Hatcher's "Algebraic Topology", in particular Corollary A.8 and Corollary A.9.

Every closed manifold is homotopy equivalent to a finite CW-complex. For a proof see Milnor's book Morse Theory in the section titled "Homotopy Type" (pg. 12 in the ancient edition I have).

(The statement of this is a remark at the end of part 1 section 3)