Wedderburn polynomials over division rings, II

TL;DR

This paper extends the study of Wedderburn polynomials over division rings, exploring their applications in matrix triangulation, diagonalization, eigenvalues, and generalizing Wedderburn's theorem through $G$-algebraic sets.

Contribution

It introduces new applications of Wedderburn polynomials to matrix theory and generalizes Wedderburn's theorem using $G$-algebraic sets in division rings.

Findings

Characterization of Wedderburn polynomials in the context of $(S,D)$-pseudo-linear transformations

Applications to matrix triangulation, diagonalization, and eigenvalues over division rings

Introduction of $G$-algebraic sets and their role in polynomial factorization

Abstract

A polynomial $f(t)$ in an Ore extension $K[t;S,D]$ over a division ring $K$ is a Wedderburn polynomial if $f(t)$ is monic and is the minimal polynomial of an algebraic subset of $K$. These polynomials have been studied in "Wedderburn polynomials over division rings,I (Journal of Pure and Applied Algebra, Vol. 186, (2004), 43-76). In this paper, we continue this study and give some applications to triangulation, diagonalization and eigenvalues of matrices over a division ring in the general setting of $(S,D)$-pseudo-linear transformations. In the last section we introduce and study the notion of $G$-algebraic sets which, in particular, permits generalization of Wedderburn's theorem relative to factorization of central polynomials.