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Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2015, Volume 157, Book 2, Pages 20–27 (Mi uzku1303)  

This article is cited in 1 scientific paper (total in 1 paper)

Determination of the minimal polynomials of algebraic numbers of the form $\operatorname{tg}^2(\pi/n)$ by the Tschirnhausen transformation

I. G. Galyautdinova, E. E. Lavrentyevab

a Povolzhskiy State University of Telecommunications and Informatics, Kazan, Russia
b Kazan (Volga Region) Federal University, Kazan, Russia

Abstract: Solutions of two problems are offered based on the Tschirnhausen transformation. The first problem is connected with the construction of minimal polynomials of the numbers of the form $\operatorname{tg}^2(\pi/n)$ by means of the Tschirnhausen transformation for all natural $n>2$. The second problem consists in finding the exact values of the roots of the equation $x^3-7x-7=0$. The solution of the problem is obtained by considering the fact that the roots of the equation produce the circular field $\mathbb Q_7$. The examples of the construction of minimal polynomials are provided.

Keywords: algebraic numbers, minimal polynomials, circular fields and subfields, Tschirnhausen transformation.

Full text: PDF file (453 kB)
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UDC: 511.61
Received: 15.12.2014

Citation: I. G. Galyautdinov, E. E. Lavrentyeva, “Determination of the minimal polynomials of algebraic numbers of the form $\operatorname{tg}^2(\pi/n)$ by the Tschirnhausen transformation”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 157, no. 2, Kazan University, Kazan, 2015, 20–27

Citation in format AMSBIB
\Bibitem{GalLav15}
\by I.~G.~Galyautdinov, E.~E.~Lavrentyeva
\paper Determination of the minimal polynomials of algebraic numbers of the form $\operatorname{tg}^2(\pi/n)$ by the Tschirnhausen transformation
\serial Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
\yr 2015
\vol 157
\issue 2
\pages 20--27
\publ Kazan University
\publaddr Kazan
\mathnet{http://mi.mathnet.ru/uzku1303}
\elib{http://elibrary.ru/item.asp?id=23819774}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. I. G. Galyautdinov, E. E. Lavrenteva, “Mnogochleny, porozhdayuschie maksimalnye veschestvennye podpolya krugovykh polei”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 158, no. 4, Izd-vo Kazanskogo un-ta, Kazan, 2016, 469–481  mathnet  elib
  • Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
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