Integral bases and monogenity of composite fields

TL;DR

This paper investigates the monogenity of high degree composite number fields, providing explicit integral bases and conditions that often prove these fields are not monogenic, especially in parametric families.

Contribution

It explicitly describes integral bases and index forms for infinite families of composite fields, establishing non-monogenity through divisibility conditions.

Findings

Most composite fields are non-monogenic due to divisibility conditions.

Explicit integral bases are constructed for fields of degrees 6 to 12.

The method avoids solving complex index form equations in high degrees.

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.