On the periodicity of continued fractions in elliptic fields

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