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. 
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/
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     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
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[1] Burbaki N., Algebra. Mnogochleny i polya. Uporyadochennye gruppy, Nauka, M., 1965, 300 pp. | MR

[2] Chebotarev N. G., Osnovy teorii Galua, ONTI GTTI, M.–L., 1934, 222 pp.

[3] Sushkevich A. K., Osnovy vysshei algebry, OGIZ, M.–L., 1941, 460 pp.

[4] Galieva L. I., Galyautdinov I. G., “Ob odnom klasse uravnenii, razreshimykh v radikalakh”, Izv. vuzov. Matem., 2011, no. 2, 22–30 | MR | Zbl

[5] Shafarevich I. O., “Novoe dokazatelstvo teoremy Kronekera–Vebera”, Trudy Matem. In-ta AN SSSR, 38, 1951, 382–387 | MR | Zbl

[6] Kostrikin A. I., Vvedenie v algebru, Ch. 3, Fizmatlit, M., 2001, 272 pp.

[7] Prasolov V. V., Mnogochleny, MTsIMO, M., 2003, 335 pp.

[8] Arnold V. I., “O klassakh kogomologii algebraicheskoi funktsii, invariantnykh otnositelno preobrazovaniya Chirngauzena”, Funktsionalnyi analiz i ego prilozheniya, 4:1 (1970), 84–85 | MR | Zbl

[9] Kolmogorov A. I., Izbrannye trudy. Matematika i mekhanika, Nauka, M., 1985, 470 pp. | MR

[10] Burbaki N., Ocherki po istorii matematiki, Izd-vo inostr. lit., M., 1963, 292 pp.