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 \(\phi (z)=z^n+c_{n-1} z^{n-1}+\cdots +c_0\) over D a “companion polynomial” \(\Phi (z)\) of degree nd with coefficients in F. The roots of \(\Phi (z)\) in D are exactly the set of conjugacy classes of the roots of \(\phi (z)\). When D is a quaternion algebra, we explain how all the roots of \(\phi (z)\) can be recovered from the roots of \(\Phi (z)\). Along the way, we are able to generalize a few known facts from \(\mathbb {H}\) to any division algebra. The first is the connection between the right eigenvalues of a matrix and the roots of its characteristic polynomial. The second is the connection between the roots of a standard polynomial and left eigenvalues of the companion matrix.