Investigating monogenity in a family of cyclic sextic fields
István Gaál
TL;DR
The paper investigates monogenity in a parametric family of cyclic sextic fields K = LM, where L is a totally real cubic field and M = Q(i) is imaginary quadratic, derived from Jones' sextic polynomials. By leveraging a relative index form I_{K/M}(γ) and a secondary factor J(γ), the authors reduce the problem to cubic Thue equations over L and a norm/linear-factor condition; they implement a Maple-based algorithm to search for generators of power integral bases across parameter values n. The main findings show monogenic instances for a small set of n in a bounded range, with explicit generators and integral bases, while a larger parameter set yields no generators within the search bound, illustrating both the method's effectiveness and its computational limits. The work provides the first non-trivial application of this approach to a parametric family of sextic fields and demonstrates practical feasibility for obtaining explicit generators in this class.
Abstract
L. Jones characterized among others monogenity of a family of cyclic sextic polynomials. Our purpose is to study monogenity of the family of corresponding sextic number fields. This also provides the first non-trivial application of the used method, emphasizing its efficiency.