Extensions of the fundamental theorem of algebra

TL;DR

This paper extends the fundamental theorem of algebra to polynomials over complex algebras, establishing eigenvalue existence for matrices with entries in such algebras, both commutative and noncommutative.

Contribution

It generalizes the fundamental theorem of algebra to complex and real algebras, including noncommutative cases, and proves eigenvalue existence for matrices over these algebras.

Findings

Eigenvalues exist for matrices over complex algebras.

Right eigenvalues exist for matrices over certain real algebras.

The results apply to both commutative and noncommutative algebra settings.

Abstract

In this paper motivated by the celebrated fundamental theorem of algebra and its standard proof utilizing Liouville's Theorem, we prove the fundamental theorem of algebra type results for both commutative and noncommutative polynomials in the setting of left (resp. right) alternative topological complex algebras whose topological duals separates their elements and that of such real algebras whose centers contain certain copies of complex numbers. An application of one of the main results of the paper is the existence of eigenvalues for matrices with entries from arbitrary finite-dimensional complex algebras. We also prove the existence of right eigenvalues for matrices with entries from finite-dimensional associative real algebras that contain copies of the complex numbers.