On the problem of periodicity of continued fraction expansions of $\sqrt{f}$ for cubic polynomials $f$ over algebraic number fields

We obtain a complete description of the fields $\mathbb K$ that are extensions of $\mathbb Q$ of degree at most $3$ and the cubic polynomials $f \in\mathbb K[x]$ such that the expansion of $\sqrt{f}$ into a continued fraction in the field of formal power series $\mathbb K((x))$ is periodic. We prove a finiteness theorem for cubic polynomials $f \in\mathbb K[x]$ with a periodic expansion of $\sqrt{f}$ for extensions of $\mathbb Q$ of degree at most $6$. We obtain a description of the periodic elements $\sqrt{f}$ for the cubic polynomials $f(x)$ defining elliptic curves with points of order $3 \le N\le 42$, $N \ne 37, 41$.Bibliography: 19 titles.

elliptic field, -units, continued fractions, periodicity, torsion points