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@MarianoSuárez-Alvarez, he was considering a certain long exact sequence from which it was not a priori clear that the corresponding maps are surjective and injective. They are, however, consider my answer to mathoverflow.net/questions/160297/…
You somehow assume that $f_5$ has a rational root, whereas in general a nonconstant polynomial can very well be irreducible over Q. Joro's example works perfectly well, since $x^2+3$ only has complex roots.
Oh, you're right. So this argument won't apply to singular cohomology, but for the spaces in question the cochain complex is quasi-isomorphic to a degreewise finitely-generated complex, so everything should work out.
