Free Fractions: An Invitation to (applied) Free Fields

TL;DR

This paper explores the construction and calculation techniques of the free field of fractions for non-commutative polynomials, enabling algebraic operations in a complex non-commutative setting.

Contribution

It introduces new methods, based on prior work, for calculating with free fractions in the free field of non-commutative rational functions.

Findings

Developed techniques for representing elements in the free field

Enabled algebraic operations in non-commutative rational functions

Built on the work of Cohn and Reutenauer

Abstract

Long before we learn to construct the field of rational numbers (out of the ring of integers) at university, we learn how to calculate with fractions at school. When it comes to "numbers", we are used to a commutative multiplication, for example 2*3=6=3*2. On the other hand --even before we can write-- we learn to talk (in a language) using words, consisting of purely non-commuting "letters" (or symbols), for example "xy" is not equal to "yx" (with the concatenation as multiplication). Now, if we combine numbers (from a field) with words (from the free monoid of an alphabet) we get non-commutative polynomials which form a ring (with "natural" addition and multiplication), namely the free associative algebra. Adding or multiplying polynomials is easy, for example (2/3*xy+z)+1/3*xy=xy+z or 2*x(yx+3*z)=2*xyx+6*xz. Although the integers and the non-commutative polynomials look rather different, they share many properties, for example the unique number of irreducible factors: x(1-yx)=x-xyx=(1-xy)x. However, the construction of the universal field of fractions (aka "free field") of the free associative algebra is highly non-trivial (but really beautiful). Therefore we provide techniques (building on the work of Cohn and Reutenauer) to calculate with free fractions (representing elements in the free field or "skew field of non-commutative rational functions") to be able to explore a fascinating non-commutative world.