A FACTORIZATION THEORY FOR SOME FREE FIELDS

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

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