Power integral bases in a family of sextic fields with quadratic subfields

Abstract

Let M = Q(i √ d) be any imaginary quadratic field with a positive square-free d. Consider the polynomialwith a parameter a ∈ Z. Let K = M (α), where α is a root of f . This is an infinite parametric family of sextic fields depending on two parameters, a and d. Applying relative Thue equations we determine the relative power integral bases of these sextic fields over their quadratic subfields. Using these results we also determine generators of (absolute) power integral bases of the sextic fields.

Monogenity in totally real extensions of imaginary quadratic fields with an application to simplest quartic fields

“…Therefore it is worthy to develop efficient methods for the resolution of special types of higher degree number fields, cf. [10], [5].…”

Monogenic Binomial Compositions

“…For any fixed degree n ≥ 2, the density of monogenic 1074 Joshua Harrington and Lenny Jones polynomials is 6/π 2 ≈ .607927 [4]. However, determining infinite families of degree-n monogenic polynomials can be difficult, and much research has been done to locate such families [1,2,5,8,10,12,13,16,17,23,29].…”

Monogenity in Totally Complex Sextic Fields, Revisited

“…However some ideas of [8] lead to a considerable improvement of that algorithm, what we are going to detail here. We also note that a parametric family of this type of number fields, consisting of composites of the simplest cubic fields and imaginary quadratic fields was studied in [10], but applying results on the connected simplest family of relative Thue equations, which counts as a more complicated approach.…”