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 Dokl. Akad. Nauk, 2018, Volume 483, Number 6, Pages 609–613 (Mi dan46857)

On the Finiteness of Hyperelliptic Fields with Special Properties and Periodic Expansion of $\sqrt f$

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

Scientific Research Institute for System Studies of RAS, Moscow

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

 Funding Agency Grant Number Russian Science Foundation 16-11-10111

DOI: https://doi.org/10.31857/S086956520003431-7

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