Extensions of the fundamental theorem of algebra

In this talk 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 several settings, e.g., the setting of associative locally convex complex algebras 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. A conjecture extending the fundamental theorem of algebra to noncommutative polynomials with coefficients from locally convex associative real algebras containing a copy of the complex numbers is proposed.