- International Electronic Journal of Algebra
- Vol: 28 Issue: 28
- A FACTORIZATION THEORY FOR SOME FREE FIELDS
A FACTORIZATION THEORY FOR SOME FREE FIELDS
Authors : Konrad Schrempf
Pages : 9-42
Doi:10.24330/ieja.768114
View : 10 | Download : 6
Publication Date : 2020-07-14
Article Type : Research
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.Keywords : Free associative algebra, factorization of non-commutative polynomials, minimal linear representation, universal field of fractions, admissible linear system, non-commutative formal power series