The following question is from Lundell’s book:
Let $X$ be a finite CW complex. For every $i$, let us suppose that $dim_{F_p} H^i (X,F_p)$ is independent of the choice of prime number $p$, where $F_p$ denotes the finite field. Prove that $H^i (X,Z)$ is free for all $i$, where $Z$ denotes the set of all integers.
Now, I know the universal coefficient theorem and cellular chain complex should be used, but I am not sure how to progress the proof. How can one prove the following using UCT and ccc.