Abstract
We obtain a complete description of cubic polynomials f over algebraic number fields \(\mathbb{K}\) of degree \(3\) over \(\mathbb{Q}\) for which the continued fraction expansion of \(\sqrt f \) in the field of formal power series \(\mathbb{K}((x))\) is periodic. We also prove a finiteness theorem for cubic polynomials \(f \in K[x]\) with a periodic expansion of \(\sqrt f \) for extensions of \(\mathbb{Q}\) of degree at most 6. Additionally, we give a complete description of such polynomials f over an arbitrary field corresponding to elliptic fields with a torsion point of order \(N \geqslant 30\).
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REFERENCES
V. P. Platonov, Russ. Math. Surv. 69 (1), 1–34 (2014).
V. P. Platonov and M. M. Petrunin, Proc. Steklov Inst. Math. 302, 354–376 (2018).
V. P. Platonov and G. V. Fedorov, Sb. Math. 209 (4), 519–559 (2018).
D. S. Kubert, Proc. London Math. Soc. (3) 33 (2), 193–237 (1976).
V. P. Platonov and G. V. Fedorov, Russ. Math. Surv. 75 (4), 785–787 (2020).
V. P. Platonov, M. M. Petrunin and, V. S. Zhgoon, Dokl. Math. 102 (1), 288–292 (2020).
V. P. Platonov, M. M. Petrunin, and Yu. N. Shteinikov, Dokl. Math. 100 (2), 440–444 (2019).
P. Parent, J. Reine Angew. Math. 1999 (506), 85–116 (1999).
M. A. Kenku and F. Momose, Nagoya Math. J. 109, 125–149 (1988).
M. Derickx, A. Etropolski, M. van Hoeij, J. S. Morrow, and D. Zureick-Brown, “Sporadic cubic torsion,” arXiv:2007.13929 (2020).
A. Sutherland, Math. Comput. 81 (278), 1131–1147 (2012).
B. Mazur, “Rational points on modular curves,” Modular Functions of One Variable V, Ed. by J. P. Serre and D. B. Zagier, Lecture Notes in Mathematics (Springer, Berlin, 1977), pp. 107–148.
D. Jeon, C. H. Kim, and E. Park, J. London Math. Soc. 74 (1), 1–12 (2006).
M. Derickx and A. Sutherland, Proc. Am. Math. Soc. 145 (10), 4233–4245 (2017).
D. Jeon, C. H. Kim, and A. Schweizer, Acta Arith. 113, 291–301 (2004).
Funding
This work was performed within the state assignment to basic scientific research, project no. 0065-2019-0011.
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Translated by I. Ruzanova
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Platonov, V.P., Petrunin, M.M. On the Finiteness of the Number of Expansions into a Continued Fraction of \(\sqrt f \) for Cubic Polynomials over Algebraic Number Fields. Dokl. Math. 102, 487–492 (2020). https://doi.org/10.1134/S1064562420060137
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DOI: https://doi.org/10.1134/S1064562420060137