  Although in general there is no meaningful concept of factorization in
fields, that in free associative algebras (over a commutative field) can be
extended to their respective free field (universal field of fractions) on the
level of minimal linear representations. We establish a factorization theory by
providing an alternative definition of left (and right) divisibility based on
the rank of an element and show that it coincides with the classical left (and
right) divisibility for non-commutative polynomials. Additionally we present an
approach to factorize elements, in particular rational formal power series,
into their (generalized) atoms. The problem is reduced to solving a system of
polynomial equations with commuting unknowns.
