  Let $X_1,\dots,X_n$ be operators in a finite von Neumann algebra and consider
their division closure in the affiliated unbounded operators. We address the
question when this division closure is a skew field (aka division ring) and
when it is the free skew field. We show that the first property is equivalent
to the strong Atiyah property and that the second property can be characterized
in terms of the non-commutative distribution of $X_1,\dots,X_n$. More
precisely, $X_1,\dots,X_n$ generate the free skew field if and only if there
exist no non-zero finite rank operators $T_1,\dots,T_n$ such that
$\sum_i[T_i,X_i]=0$. Sufficient conditions for this are the maximality of the
free entropy dimension or the existence of a dual system of $X_1,\dots,X_n$.
Our general theory is not restricted to selfadjoint operators and thus does
also include and recover the result of Linnell that the generators of the free
group give the free skew field.
  We give also consequences of our result for the question of atoms in the
distribution of rational functions in free variables or in the asymptotic
eigenvalue distribution of matrices over polynomials in asymptotically free
random matrices. This solves in particular a conjecture of Charlesworth and
Shlyakhtenko.
