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

We obtain a complete description of fields $\mathbb{K}$ that are quadratic extensions of $\mathbb{Q}$ and of cubic polynomials $f\in\mathbb{K}[x]$ for which a 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 $f\in\mathbb{K}[x]$ with a periodic expansion of $\sqrt{f}$ over cubic and quartic extensions of $\mathbb{Q}$.