A FACTORIZATION THEORY FOR SOME FREE FIELDS

Year 2020, Volume: 28 Issue: 28, 9 - 42, 14.07.2020

Konrad Schrempf

https://doi.org/10.24330/ieja.768114

Cited By: 1

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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