Existence of polynomials with given roots over non-commutative rings

TL;DR

This paper investigates the existence of polynomials with specified roots over non-commutative rings, providing conditions and criteria for their existence, especially over division rings and matrix rings, with explicit examples.

Contribution

It establishes the existence of polynomials with given roots over division rings and provides sufficient conditions for general associative rings, including criteria for matrix rings.

Findings

Existence of degree n polynomials with given roots over division rings.

Sufficient conditions for polynomials with specified roots over general rings.

Criteria for second-degree polynomials with given roots over matrix rings.

Abstract

The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there exists a polynomial of degree n whose roots are these elements. Sufficient conditions for the existence of such a polynomial are also obtained in the case of an arbitrary (not necessarily division) associative ring with identity. The case of polynomials defined over a matrix ring over a field is considered separately; for such polynomials a criterion for the existence of a second-degree polynomial with given roots is obtained; examples of constructing polynomials with given roots are also given.