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

Question

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 $$F_{\alpha \beta}~=~\partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}~? $$

Answer 1

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.