# Linearizing the Word Problem in (some) Free Fields

## TL;DR

This paper presents a linear algebra-based method for solving the word problem in free fields using minimal linear representations, enabling rational identity testing and inverse construction.

## Contribution

It introduces a linear algebra approach leveraging Cohn and Reutenauer's normal form for free fields, with a new method for minimal linear representation construction.

## Key findings

- Effective solution to the word problem in free fields.
- Method for testing rational identities.
- Construction of minimal linear representations for inverses.

## Abstract

We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover we provide a construction of minimal linear representations for the inverse of non-zero elements.