danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2024-68-5-359-364dan-1210Research ArticleМАТЕМАТИКАMATHEMATICSКорни многочленов с коэффициентами в кольцах с делениемRoots of polynomials over division ringsГуторА. Г.GoutorA. G.

Гутор Алина Геннадьевна – ст. преподаватель.

пр. Независимости, 4, 220030, Минск

Alina G. Goutor – Senior Lecturer, Belarusian State University.

4, Nezavisimosti Ave., 220030, Minsk

goutor@bsu.byhttps://orcid.org/0000-0002-2597-7964ТихоновС. В.TikhonovS. V.

Тихонов Сергей Викторович – канд. физ.-мат. наук, заведующий кафедрой.

пр. Независимости, 4, 220030, Минск

Sergey V. Tikhonov – Ph. D. (Physics and Mathematics), Head of the Department, Belarusian State University.

4, Nezavisimosti Ave., 220030, Minsk

tikhonovsv@bsu.byБелорусский государственный университетBelarusian State University202406112024685359364Copyright © Гутор А.Г., Тихонов С.В., 20242024Гутор А.Г., Тихонов С.В.Goutor A.G., Tikhonov S.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/1210

В работе изучены свойства многочленов с коэффициентами в кольцах с делением. Получены формулы для нахождения корней многочленов, являющихся произведением линейных множителей, обобщающие известные результаты для кватернионных алгебр. Как известно, если минимальный многочлен класса сопряженности А в некоммутативном кольце с делением является квадратичным, то любой многочлен, имеющий два корня в A, обнуляется тождественно на A. В работе показано, что в случае класса сопряженности с минимальным многочленом большей степени ситуация принципиально другая. Для любого класса сопряженности с минимальным многочленом степени >2 построен квадратичный многочлен, имеющий бесконечно много корней в этом классе, при этом в данном классе сопряженности имеется бесконечно много элементов, не являющихся корнями такого многочлена.

In this article, we study the properties of polynomials over division rings. Formulas for finding roots of polynomials which are the products of linear factors are obtained. These formulas generalize the known results for quaternion algebras. As known, if a minimal polynomial of a conjugacy class A in a noncommutative division ring is quadratic, then any polynomial having two roots in A vanishes identically on A. We show that in the case of a conjugacy class with minimal polynomial of larger degree, the situation is completely different. For any conjugacy class with minimal polynomial of degree >2, we construct a quadratic polynomial with infinitely many roots in this class, but there also are infinitely many elements in this class which are not the roots of this polynomial.

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The authors declare that there are no conflicts of interest present.