Non-monogenity in a family of octic fields

TL;DR

This paper proves that a specific family of Galois octic fields, constructed from square-free integers, do not admit power integral bases, using a novel congruence-based method applicable to parametric families.

Contribution

It introduces a new approach to determine non-monogenity in algebraic number fields through congruence considerations, applicable to parametric families.

Findings

The fields are non-monogene for the specified parameters.

A new congruence-based method for analyzing monogenity.

Explicit construction of integral bases in the family.

Abstract

Let $m$ be a square-free positive integer, $m\equiv 2,3 \; (\bmod \; 4)$. We show that the number field $K=Q(i,\sqrt[4]{m})$ is non-monogene, that is it does not admit any power integral bases of type $\{1,\alpha,\ldots,\alpha^7\}$. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using only congruence considerations. Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields. It is well applicable in parametric families of number fields. We calculate the index of elements as polynomials depending on the parameter, factor these polynomials and consider systems of congruences according to the factors.