On the Problem of Periodicity of Continued Fraction Expansions of $$\sqrt f $$for Cubic Polynomials over Number Fields

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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 $$\mathbb{Q}$$ .