electromagnetism - Decomposition of a vectorial field in free-curl and free-divergence fields

Is it always possible to do that decomposition? I'm asking it because Helmholtz theorem says a field on $\mathbb{R}^3$ that vanishes at infinity ($r\to \infty$) can be decomposed univocally into a gradient and a curl. But I also know, for example, that a constant field $\mathbf{E}$ on $\mathbb{R}^3$ is a gradient (not univocally definied): $\mathbf{E}(x+y+z+\mbox{constant})$. And the electric field is $-\nabla G+ d\mathbf{A}/dt$, where $\mathbf{A}$ can be (Coulomb Gauge) free-divergence.

So, is it always possible to do the decomposition of a (regular, of course) field on $\mathbb{R}^3$ into two fields, free-curl and free-divergence? And on a limited domain?