DETERMINING ELEMENTS OF MINIMAL INDEX IN AN INFINITE FAMILY OF TOTALLY REAL BICYCLIC BIQUADRATIC NUMBER FIELDS

Abstract

Let c ≠ 2 be a positive integer such that c and c + 4 are square-free. We consider the infinite parametric family of bicyclic biquadratic fields K = Q(sqrt {; ; 2c}; ; , sqrt {; ; 2(c + 4)}; ; ). We determine the integral basis of the field. We show that K admits no power integral basis, determine the minimal index and all elements of minimal index. We use the solutions of a parametric family of quartic Thue equations and extensive numerical calculations by Maple and Magma are also involved.