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.
Highlights
From an algebraic point of view fields are usually not very interesting due to the lack of “structure”, for example, they do not have non-zero non-units
The main idea is to view all elements in terms of their normal form [9]
In [22] we showed that in the free associative algebra there is a rather natural correspondence between a factorization of an element and blocks of zeros in its minimal linear representations
Summary
From an algebraic point of view fields are usually not very interesting (with respect to factorization) due to the lack of “structure”, for example, they do not have non-zero non-units. The main idea is to view all elements in terms of their normal form (minimal linear representation) [9]. In [22] we showed that in the free associative algebra there is a rather natural correspondence between a factorization of an element and (upper right) blocks of zeros in (a special form of) its minimal linear representations. Since each non-zero element (in the free field) is invertible, we can use both, its rank and that of its inverse, for example, the inverse of a polynomial of rank n ≥ 2 has rank n − 1. This exposition is not meant to serve as an introduction, neither to free fields nor to non-commutative factorization (in free associative algebras).
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