A note on polynomial equations over algebras

TL;DR

This paper establishes conditions for solutions to polynomial equations over various algebras, generalizes the fundamental theorem of algebra for quaternions, and explores the limits of such generalizations.

Contribution

It provides new sufficient conditions for polynomial solvability over general algebras and extends the fundamental theorem of algebra to certain quaternion polynomials.

Findings

Conditions for solutions over real and complex algebras

Generalization of the fundamental theorem of algebra for quaternions with two monomials

Failure of the theorem for polynomials with three monomials

Abstract

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the fundamental theorem of algebra for quaternions to polynomials with two monomials in the leading form, while showing that it fails for three.