Proof that 4-potential exists from Gauss-Faraday field equation

This is a problem concerning covariant formulation of electromagnetism.

Given $$\partial_{[\alpha} F_{\beta\gamma]}~=~ 0$$ how does one prove that $F$ can be obtained from a 4-potential $A$ such that $$F_{\alpha \beta}~=~\partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}~?$$

The local existence of a one-form $$A$$ such that the closed two-form $$F$$ is exact $$F=\mathrm{d}A$$ is a consequence of the Poincare Lemma. There might be global obstructions.
• Great answer! Do you know if there is an extension of the Poincare Lemma for the case where the space where $F$ is defined is not an open subset of $\mathbb{R}^{n}$? I mean, you may have a non-compact manifold that cannot be embedded. Just curious – Arthur Suvorov May 20 '14 at 23:12
• Well, a sufficient condition for the global existence of $A\in\Gamma(T^{*}M)$ on a manifold $M$ is if the second cohomology group $H^2(M)=0$ is trivial, but you probably know that already. – Qmechanic May 21 '14 at 18:38