  We investigate compact complex manifolds of dimension three and second Betti
number $b_2(X) = 0$. We are interested in the algebraic dimension $a(X)$, which
is by definition the transcendence degree of the field of meromorphic functions
over the field of complex numbers. The topological Euler characteristic
$\chi_{\mathrm{ top}}(X) $ equals the third Chern class $c_3(X)$ by a theorem
of Hopf. Our main result is that, if $X$ is a compact 3-dimensional complex
manifold with $b_2(X) = 0$ and $a(X) > 0$, then $c_3(X) = \chi_{\rm top}(X) =
0$, that is, we either have $b_1(X) = 0, \ b_3(X) = 2$ or $b_1(X) = 1, \ b_3(X)
= 0.$
