A FACTORIZATION THEORY FOR SOME FREE FIELDS

Authors : Konrad SCHREMPF

Pages : 9-42

Doi:10.24330/ieja.768114

View : 19 | Download : 14

Publication Date : 2020-07-14

Article Type : Research Paper

Abstract :Although in general there is no meaningful concept of factorization in fields, that in free associative algebras insert ignore into journalissuearticles values(over a commutative field); can be extended to their respective free field insert ignore into journalissuearticles values(universal field of fractions); on the level of minimal linear representations . We establish a factorization theory by providing an alternative definition of left insert ignore into journalissuearticles values(and right); divisibility based on the rank of an element and show that it coincides with the ` classical`` left insert ignore into journalissuearticles values(and right); divisibility for non-commutative polynomials. Additionally we present an approach to factorize elements, in particular rational formal power series, into their insert ignore into journalissuearticles values(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