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arXiv · 2501.02897

Existence of polynomials with given roots over non-commutative rings

Alina G. Goutor
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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.

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Alina G. Goutor. 2025-01-06. Existence of polynomials with given roots over non-commutative rings. https://arxiv.org/abs/2501.02897

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