Standard Polynomial Equations over Division Algebras

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.