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

We obtain a complete description of cubic polynomials f over algebraic number fields of degree over for which the continued fraction expansion of in the field of formal power series is periodic. We also prove a finiteness theorem for cubic polynomials with a periodic expansion of for extensions of 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 .