On the finiteness of the number of expansions into a continued fraction of $\sqrt f$ for cubic polynomials over algebraic number fields

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 $N\ge30$.