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Extensions of the fundamental theorem of algebra

Bamdad R. Yahaghi

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.

Extensions of the fundamental theorem of algebra

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.
Paper Structure (2 sections, 7 theorems, 24 equations)

This paper contains 2 sections, 7 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Theorem 1.1

(i) Let $\mathbb{A}$ be a finite-dimensional real or complex algebra and $||.||$ be a vector space norm on $\mathbb{A}$. Then, there exists an $M > 0$ such that $M||.||$ is an algebra norm on $\mathbb{A}$. Therefore, every finite-dimensional real or complex algebra can be normed. Moreover, if a real

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5