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Diophantine Equation Generated by the Maximal Subfield of a Circular Field

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Abstract

Using the fundamental basis of the field \(L_9=\mathbb{Q} (2\cos(\pi/9)),\) the form \(N_{L_9}(\gamma)=f(x, y, z)\) is found and the Diophantine equation \(f(x,y,z)=a\) is solved. A similar scheme is used to construct the form \(N_{L_7}(\gamma)=g(x,y,z)\). The Diophantine equation \(g (x, y, z)=a\) is solved.

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ACKNOWLEDGMENT

We are grateful to the seminar of the Chair of Algebra and Mathematical Logics of Kazan Federal University under the supervision of Professor M.M. Arslanov, for the possibility of making a report there with the content of this paper. We express our thankfulness to the members of this seminar for their constructive remarks. Our deep gratitude is to A.N. Abysov for the attention to our work and for the invaluable consultation.

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Authors and Affiliations

  1. Tatar State University of Humanities and Education, 2 Tatarstan str., 420021, Kazan, Russia

    I. G. Galyautdinov

  2. Kazan Federal University, 18 Kremlyovskaya str., 420008, Kazan, Russia

    E. E. Lavrentyeva

Authors
  1. I. G. Galyautdinov
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  2. E. E. Lavrentyeva
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Correspondence to E. E. Lavrentyeva.

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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 7, pp. 45–55.

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Galyautdinov, I.G., Lavrentyeva, E.E. Diophantine Equation Generated by the Maximal Subfield of a Circular Field. Russ Math. 64, 38–47 (2020). https://doi.org/10.3103/S1066369X20070051

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  • DOI: https://doi.org/10.3103/S1066369X20070051

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