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We prove that for "most" closed 3-dimensional manifolds $M$, the existence ofa closed non singular one-form in a given cohomology class $u\in H^1 (M,\bf R)$is equivalent to the fact that every twisted Alexander polynomial$\Delta^H(M,u) \in {\bf Z}[G/\ker u]$ associated to a normal subgroup withfinite index $H < \pi_1(M)$ has a unitary $u$-minimal term.

Fibered Cohomology Classes in Dimension Three, Twisted Alexander Polynomials and Novikov Homology