The existence of polynomials with given roots over non-commutative rings

Authors

  • Alina G. Goutor Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus

Keywords:

ring, division ring, polynomial, ring of square matrices
Supporting Agencies
The author express gratitude to PhD (physics and mathematics), docent S. V. Tikhonov for interesting ideas and useful comments.

Abstract

This paper studies the problem of the existence of polynomials with given roots over associative non-commutative rings. It is shown that for arbitrary n elements of an associative division ring there exists a polynomial of degree n whose roots are these elements. The sufficient conditions for the existence of such a polynomial for elements of an arbitrary (not necessarily division) associative ring with unity are determined. For polynomials defined over a ring of square matrices over a field, a criterion for the existence of a second-degree polynomial with given roots is obtained, and examples of constructing polynomials with given roots are given.

Author Biography

  • Alina G. Goutor, Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus

    senior lecturer at the department of intelligent modelling methods, faculty of mechanics and mathematics

References

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Published

2025-05-08

Issue

No. 1 (2025): Journal of the Belarusian State University. Mathematics and Informatics

Section

Mathematical Logic, Algebra and Number Theory

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How to Cite

[1]
Goutor, A.G. 2025. The existence of polynomials with given roots over non-commutative rings. Journal of the Belarusian State University. Mathematics and Informatics. 1 (May 2025), 6–13.