On the classification problem for polynomials $f$ with a periodic continued fraction expansion of $\sqrt{f}$ in hyperelliptic fields

The classical problem of the periodicity of continued fractions for elements of hyperelliptic fields has a long and deep history. This problem has up to now been far from completely solved. A surprising result was obtained in [1] for quadratic extensions defined by cubic polynomials with coefficients in the field $\mathbb{Q}$ of rational numbers: except for trivial cases there are only three (up to equivalence) cubic polynomials over $\mathbb{Q}$ whose square root has a periodic continued fraction expansion in the field $\mathbb{Q}((x))$ of formal power series. In view of the results in [1], we completely solve the classification problem for polynomials $f$ with periodic continued fraction expansion of $\sqrt{f}$ in elliptic fields with the field of rational numbers as the field of constants.