A Banach Algebraic Version Of The Fundamental Theorem of Algebra

TL;DR

This paper extends the Fundamental Theorem of Algebra to polynomials with coefficients in a unital complex Banach algebra, showing such polynomials always have a point where they are not invertible.

Contribution

It introduces a Banach algebraic version of the Fundamental Theorem of Algebra, generalizing classical results to a broader algebraic context.

Findings

Existence of a point where p(z) is not invertible for monic polynomials in Banach algebras

Generalization of the classical theorem to complex Banach algebra coefficients

Provides a new perspective on polynomial roots in algebraic structures

Abstract

For a monic polynomial p(z) with coefficients in a unital complex Banach algebra, we prove that there exist a complex number z such that p(z)is not invertible