On the Periodicity of Continued Fractions in Elliptic Fields
Academician of the Russian Academy of Sciences V. P. Platonov* and G. V. Fedorov**
Translated by O. Sipacheva
Scientific Research Institute for System Analysis, Russian Academy of Sciences, Moscow, 117218 Russia
Correspondence to: * e-mail: platonov@niisi.ras.ru
Correspondence to: ** e-mail: glebonyat@mail.ru
Received 30 March, 2017
Abstract—Article [1] raised the question of the finiteness of the number of square-free polynomials $f \in \mathbb{Q}[h]$ of fixed degree for which $\sqrt f $ has periodic continued fraction expansion in the field $\mathbb{Q}((h))$ and the fields $\mathbb{Q}(h)(\sqrt f )$ are not isomorphic to one another and to fields of the form $\mathbb{Q}(h)(\sqrt {c{{h}^{n}} + 1} )$, where $c \in \mathbb{Q}{\text{*}}$ and $n \in \mathbb{N}$. In this paper, we give a positive answer to this question for an elliptic field $\mathbb{Q}(h)(\sqrt f )$ in the case $degf = 3$.
DOI: 10.1134/S1064562417040068