Algebraic Conjugacy Classes and Skew Polynomial Rings

Abstract

The goal of this paper is to develop further the theory of skew polynomial rings over division rings, using as our main tools the notions of invariant and semi-invariant polynomials. These notions arise naturally when one tries to study the algebraic conjugacy classes (in a suitably generalized sense) of the underlying division ring. A substantial part of our effort will also be devoted to the investigation of the properties and the characterizations of algebraic derivations, algebraic endomorphisms, and their respective minimal polynomials. This investigation is made possible by the discovery of the relationship between polynomial equations and differential equations, and the relationship between polynomial dependence and linear dependence. Applications of these results to the study of non-commu tative Hilbert 90-type theorems will be presented in a forthcoming work [LL2].