Standard Polynomial Equations over Division Algebras

Abstract

Given a central division algebra $D$ of degree $d$ over a field $F$, we associate to any standard polynomial $\ddot{I}•(z)=z^n+c_{n-1}{z}^{n-1}+\cdots +c_0$ over $D$ a "companion polynomial" $\hat{I}¦(z)$ of degree $nd$ with coefficients in $F$ whose roots are exactly the conjugacy classes of the roots of $\ddot{I}•(z)$. We explain how in case $D$ is a quaternion algebra, all the roots of $\ddot{I}•(z)$ can be recovered from the roots of $\hat{I}¦(z)$. 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.

9 pages