Standard Polynomial Equations over Division Algebras

TL;DR

This paper introduces a method to associate a companion polynomial over a base field to standard polynomials over division algebras, enabling root recovery and generalizing eigenvalue theorems beyond quaternions.

Contribution

It generalizes known quaternion algebra results to arbitrary division algebras, linking roots of polynomials to eigenvalues and introducing a companion polynomial construction.

Findings

Roots of the companion polynomial determine roots of the original polynomial.

The method allows root recovery in quaternion and general division algebra cases.

Generalization of eigenvalue-root relationships to all division algebras.

Abstract

Given a central division algebra $D$ of degree $d$ over a field $F$, we associate to any standard polynomial $\phi(z)=z^n+c_{n-1} z^{n-1}+\dots+c_0$ over $D$ a "companion polynomial" $\Phi(z)$ of degree $n d$ with coefficients in $F$ whose roots are exactly the conjugacy classes of the roots of $\phi(z)$. We explain how in case $D$ is a quaternion algebra, all the roots of $\phi(z)$ can be recovered from the roots of $\Phi(z)$. On the way, we also generalize certain theorems that were known for $\mathbb{H}$ to any division algebra, such as the connection between the right eigenvalues of a matrix and the roots of its characteristic polynomial, and the connection between the roots of a standard polynomial and left eigenvalues of the companion matrix.