Existence of polynomials with given roots over non-commutative 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.