Determination of the Minimal Polynomials of Algebraic Numbers of the Form tg2 (π/n) by the Tschirnhausen Transformation
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 tg2 (π/n) by means of the Tschirnhausen transformation for all natural n > 2. The second problem consists in find...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2015-06-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | https://kpfu.ru/portal/docs/F1145570715/157_2_phys_mat_2.pdf |
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| author | I.G. Galyautdinov E.E. Lavrentyeva |
| author_facet | I.G. Galyautdinov E.E. Lavrentyeva |
| author_sort | I.G. Galyautdinov |
| collection | DOAJ |
| description | 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 tg2 (π/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 x3 − 7x − 7 = 0. The solution of the problem is obtained by considering the fact that the roots of the equation produce the circular field Q7 . The examples of the construction of minimal polynomials are provided. |
| format | Article |
| id | doaj-art-56a6b0d463b241f7a04dec5f77286cc1 |
| institution | Kabale University |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2015-06-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-56a6b0d463b241f7a04dec5f77286cc12025-01-02T11:19:31ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982015-06-0115722027Determination of the Minimal Polynomials of Algebraic Numbers of the Form tg2 (π/n) by the Tschirnhausen TransformationI.G. Galyautdinov0E.E. Lavrentyeva1 Volga Region State University of Telecommunications and Informatics, Kazan, 420000Kazan Federal University, Kazan, 420008 RussiaSolutions 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 tg2 (π/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 x3 − 7x − 7 = 0. The solution of the problem is obtained by considering the fact that the roots of the equation produce the circular field Q7 . The examples of the construction of minimal polynomials are provided.https://kpfu.ru/portal/docs/F1145570715/157_2_phys_mat_2.pdfalgebraic numbersminimal polynomialscircular fields and subfieldstschirnhausen transformation |
| spellingShingle | I.G. Galyautdinov E.E. Lavrentyeva Determination of the Minimal Polynomials of Algebraic Numbers of the Form tg2 (π/n) by the Tschirnhausen Transformation Учёные записки Казанского университета: Серия Физико-математические науки algebraic numbers minimal polynomials circular fields and subfields tschirnhausen transformation |
| title | Determination of the Minimal Polynomials of Algebraic Numbers of the Form tg2 (π/n) by the Tschirnhausen Transformation |
| title_full | Determination of the Minimal Polynomials of Algebraic Numbers of the Form tg2 (π/n) by the Tschirnhausen Transformation |
| title_fullStr | Determination of the Minimal Polynomials of Algebraic Numbers of the Form tg2 (π/n) by the Tschirnhausen Transformation |
| title_full_unstemmed | Determination of the Minimal Polynomials of Algebraic Numbers of the Form tg2 (π/n) by the Tschirnhausen Transformation |
| title_short | Determination of the Minimal Polynomials of Algebraic Numbers of the Form tg2 (π/n) by the Tschirnhausen Transformation |
| title_sort | determination of the minimal polynomials of algebraic numbers of the form tg2 π n by the tschirnhausen transformation |
| topic | algebraic numbers minimal polynomials circular fields and subfields tschirnhausen transformation |
| url | https://kpfu.ru/portal/docs/F1145570715/157_2_phys_mat_2.pdf |
| work_keys_str_mv | AT iggalyautdinov determinationoftheminimalpolynomialsofalgebraicnumbersoftheformtg2pnbythetschirnhausentransformation AT eelavrentyeva determinationoftheminimalpolynomialsofalgebraicnumbersoftheformtg2pnbythetschirnhausentransformation |