Geodesic
Determination of the minimal polynomials of algebraic numbers of the form $\operatorname{tg}^2(\pi/n)$ by the Tschirnhausen transformation
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 157 (2015) no. 2, pp. 20-27

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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
Mots-clés : Tschirnhausen transformation. 
@article{UZKU_2015_157_2_a1,
     author = {I. G. Galyautdinov and E. E. Lavrentyeva},
     title = {Determination of the minimal polynomials of algebraic numbers of the form $\operatorname{tg}^2(\pi/n)$ by the {Tschirnhausen} transformation},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {20--27},
     publisher = {mathdoc},
     volume = {157},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2015_157_2_a1/}
}
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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. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 157 (2015) no. 2, pp. 20-27. http://geodesic.mathdoc.fr/item/UZKU_2015_157_2_a1/