From electrodynamics we know that $\boldsymbol{\nabla}\mathbf{B}=\mathbf 0$ hence we can introduce a vector potential such that $\mathbf{B}=[\boldsymbol \nabla\times \mathbf{A}]$.
What is the general mathematical proof that a vector field with zero divergence can be expressed as a curl of another vector field (that this field would necessarily exist)? What are the restrictions on the fields?
I suspect this question was asked before, I just didn't manage to find a clean answer.
The framework of vector-analysis provides certain concepts and identities regarding how 'vectors' can be manipulated.
One of them being: a divergence-less $[\nabla.\vec{X}=0]$ vector field should wind upon itself, or simply be solenoidal $ [\vec{X} \text{ is curl of some other field}\implies \vec{X}=\nabla\times \vec{Y}]$ since $\forall \vec{Y}\,\nabla . (\nabla \times \vec{Y})=0$. (Unlike many other identities in vector analysis, this one's quite easy to visualize.)
And since EM deals with vector fields representing E and B in space, the divergence-lessness of $\vec{B}$ gives its solenoidal character, being $\vec{B}=\nabla \times \vec{A}$
But as @symplectomorphic mentioned in the comment, the restrictions are quite intricate indeed, in general.