A Factorization Theory for some Free Fields

TL;DR

This paper develops a factorization theory for free fields, extending concepts of divisibility and atomization from non-commutative polynomials to rational formal power series, using minimal linear representations.

Contribution

It introduces an alternative divisibility definition based on rank, aligning with classical divisibility, and provides a method to factorize elements into atoms via polynomial equations.

Findings

Established a factorization framework for free fields.

Connected divisibility in free fields to minimal linear representations.

Provided an approach to factorize rational formal power series.

Abstract

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.