On the Periodicity of Continued Fractions in Hyperelliptic Fields over Quadratic Constant Field

Abstract

We give a description of the cubic polynomials f(x) with coefficients in the quadratic number fields \(\mathbb{Q}(\sqrt{5})\) and \(\mathbb{Q}(\sqrt{-15})\) for which the continued fraction expansion of the irrationality \(\sqrt {f\left( x \right)} \) is periodic.