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.