Power integral bases in a family of octic fields

TL;DR

This paper investigates the existence of multiple generators of power integral bases in a specific family of octic number fields, extending previous research and building on recent findings by L. Jones.

Contribution

It extends the study of power integral bases to a new family of octic fields, exploring whether these fields admit generators beyond the root of the defining polynomial.

Findings

Identified conditions for multiple power integral bases in the family

Extended previous results to a broader class of octic fields

Provided new examples of non-unique power integral bases

Abstract

Several recent results prove the monogenity of some polynomials. In these cases the root of the polynomial generates a power integral basis in the number field generated by the root. A straightforward question is whether such a number field admits other generators of power integral bases? We have investigated this problem in some previous papers and here we extend this research to a family of octic polynomials, following a recent result of L. Jones.