Integral Bases and Monogenity of Composite Fields

Abstract

We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually can not hold, which implies the non-monogenity of the fields.The fields that we consider are higher degree number fields, of degrees 6 up to 12. The non-monogenity of the number fields is stated very often as a consequence of the non-existence of the solutions of the index form equation. Up to our knowlegde it is not at all feasible to solve the index form equation in these high degree fields, especially not in a parametric form.On the other hand our method implies directly the non-monogenity in almost all cases. We obtain our results in a parametric form, characterizing these infinite parametric families of composite fields.

The scheme of monogenic generators I: representability

“…Specializing families of polynomials to obtain monogenic extensions is investigated in [49]. In large part due to the group in Debrecen, there is a vast literature involving relative monogenicity: [23,26,[28][29][30]38,39], and [27].…”

The Scheme of Monogenic Generators I: Representability

“…Specializing families of polynomials to obtain monogenic extensions is investigated in [Kön18]. In large part due to the group in Debrecen, there is a vast literature involving relative monogeneity: [Győ80], [Győ81], [Gaá01], [GP00], [GS13], [GRS16], [GR19b], and [GR19a].…”

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

“…Remark. Note that in [11] we obtained conditions on the monogenity of composites of fields, among others of simplest quartic fields and quadratic fields. We did not assume that the discriminants are relative prime and involved also real quadratic fields.…”

“…In some cases we considered monogenity in composites of fields, see [4], [11]. In these cases the index form factorizes that makes the resolution of the index form equation easier.…”