matatecsМатематика и теоретические компьютерные наукиMathematics and Theoretical Computer Science2949-3919Казанский (Приволжский) федеральный университет10.26907/2949-3919.2026.2.4-75matatecs-109Research ArticleСТАТЬИКвадратичный закон взаимности и его обобщенияThe quadratic reciprocity law and its generalizationsАбызовА. Н.AbyzovA. N.

Адель Наилевич Абызов,

ул. Кремлевская, д. 18, г. Казань, 420008.

Adel Nailevich Abyzov,

18, Kremlyovskaya str., Kazan 420008.

Adel.Abyzov@kpfu.ruБуутайП. Н.BuutaiP. N.

Павел Николаевич Буутай,

ул. Мясницкая, д. 20, г. Москва, 101000;

ул. Белинского, д. 58, г. Якутск, 677000.

Pavel Nikolaevich Buutai,

20, Myasnitskaya str., Moscow 101000;

58, Belinskogo str., Yakutsk 677000.

buutay.p@hse.ruКазанский (Приволжский) федеральный университетРоссияKazan Federal UniversityRussian FederationНациональный исследовательский университет «Высшая школа экономики»; Северо-Восточный федеральный университет имени М.К. АммосоваРоссияHSE University; North-Eastern Federal UniversityRussian Federation20262907202642475Copyright © Абызов А.Н., Буутай П.Н., 20262026Абызов А.Н., Буутай П.Н.Abyzov A.N., Buutai P.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matatecs.elpub.ru/jour/article/view/109

Статья носит обзорно-методический характер и посвящена развитию идей Е.И. Золотарёва, заложенных в его подходе к доказательству квадратичного закона взаимности (1872 г.). Мы рассматриваем расширения подхода Золотарёва на абстрактные числовые кольца, приведенные в работе А. Бруньята и П.Л. Кларка (2015 г.), и на конечные группы, изученные в статье У. Дьюка и К. Хопкинс (2005 г.). Также обсуждаются связи этих исследований с некоторыми классическими результатами и различными подходами к доказательству квадратичного закона взаимности.

This paper is expository and methodological in nature and is devoted to the development of E.I. Zolotarev’s ideas embedded in his approach to the proof of the quadratic reciprocity law (1872). We consider extensions of Zolotarev’s approach to abstract number rings presented in the work of A. Brunyate and P.L. Clark (2015), and to finite groups studied in the paper by W. Duke and K. Hopkins (2005). We also discuss the connections between these studies and certain classical results, as well as various approaches to proving the quadratic reciprocity law. 

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