On the Finiteness of the Number of Expansions into a Continued Fraction of $$\sqrt f $$ for Cubic Polynomials over Algebraic Number Fields

Abstract

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 $$N \geqslant 30$$ .