On the periodicity of continued fractions in elliptic fields

Abstract

Article [1] raised the question of the finiteness of the number of square-free polynomials f ∈ ℚ[h] of fixed degree for which $\sqrt f$ has periodic continued fraction expansion in the field ℚ((h)) and the fields ℚ(h)($\sqrt f$) are not isomorphic to one another and to fields of the form ℚ(h)$\left( {\sqrt {c{h^n} + 1} } \right)$, where c ∈ ℚ* and n ∈ ℕ. In this paper, we give a positive answer to this question for an elliptic field ℚ(h)($\sqrt f$) in the case deg f = 3.