Extensions of the fundamental theorem of algebra

In this talk motivated by the celebrated fundamental theorem of algebra and itsstandard proof utilizing Liouville’s Theorem, we prove the fundamental theorem ofalgebra type results for both commutative and noncommutative polynomials in severalsettings, e.g., the setting of associative locally convex complex algebras and thatof such real algebras whose centers contain certain copies of complex numbers. Anapplication of one of the main results of the paper is the existence of eigenvalues formatrices with entries from arbitrary finite-dimensional complex algebras. A conjectureextending the fundamental theorem of algebra to noncommutative polynomialswith coefficients from locally convex associative real algebras containing a copy of thecomplex numbers is proposed.

کلیدواژه ها:

(Noncommutative) Polynomials ، Real/Complex algebras ، Commutative/Alternative/ Associative/Noncommutative/Nonassociative algebras ، (Left/Right) eigenvalues ، Singular elements ، Copies of complex numbers.