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

A Factorization Theory for some Free Fields