On splitting polynomials with noncommutative coefficients

TL;DR

This paper explores the properties of splitting polynomials with noncommutative coefficients, demonstrating invariance under cyclic permutations and linking roots to the polynomial's structure, with insights from Galois theory.

Contribution

It establishes that cyclic permutations of linear factors yield the same product and that all roots commute with coefficients, extending classical polynomial factorization to noncommutative settings.

Findings

Cyclic permutations of linear factors produce identical results.

All roots of the polynomial commute with the coefficients.

Connections to Galois theory are analyzed.

Abstract

It is shown that for every splitting of a polynomial with noncommutative coefficients into linear factors $(X-a_{k})$ with $a_{k}$'s commuting with coefficients, any cyclic permutation of linear factors gives the same result and all $a_{k}$ are roots of that polynomial. Examples are given and analyzed from Galois theory point of view.