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On the Finiteness of Hyperelliptic Fields with Special Properties and Periodic Expansion of √f

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Abstract

We prove the finiteness of the set of square-free polynomials f ∈ k[x] of odd degree distinct from 11 considered up to a natural equivalence relation for which the continued fraction expansion of the irrationality \(\sqrt {f\left( x \right)} \) in k((x)) is periodic and the corresponding hyperelliptic field k(x)(√f) contains an S-unit of degree 11. Moreover, it was proved for k = ℚ that there are no polynomials of odd degree distinct from 9 and 11 satisfying the conditions mentioned above.

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Authors and Affiliations

  1. Scientific Research Institute for System Analysis, Russian Academy of Sciences, Moscow, 117218, Russia

    V. P. Platonov, V. S. Zhgoon, M. M. Petrunin & Yu. N. Shteinikov

Authors
  1. V. P. Platonov
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  2. V. S. Zhgoon
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  3. M. M. Petrunin
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  4. Yu. N. Shteinikov
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Corresponding author

Correspondence to V. P. Platonov.

Additional information

Original Russian Text © V.P. Platonov, V.S. Zhgoon, M.M. Petrunin, Yu.N. Shteinikov, 2018, published in Doklady Akademii Nauk, 2018, Vol. 483, No. 6.

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Platonov, V.P., Zhgoon, V.S., Petrunin, M.M. et al. On the Finiteness of Hyperelliptic Fields with Special Properties and Periodic Expansion of √f. Dokl. Math. 98, 641–645 (2018). https://doi.org/10.1134/S1064562418070281

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  • DOI: https://doi.org/10.1134/S1064562418070281

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