I am stuck on the following problem from Lundell’s “Topology of CW Complexes”.
Let $X$ be a finite CW-Complex. For every $i$, suppose that $dim_{F_p} H^i(X,F_p)$ is independent of the choice of the prime number $p$ ($F_p$ denotes the finite field). Prove that $H^i (X,\mathbb{Z})$ are free for all $i$, where $\mathbb{Z}$ is the set of all integers.
My problem is that I do not know how to do that considering the finite field. Assuming that condition above, how can I prove that $H^i (X,\mathbb{Z})$ are free groups deriving from that?