On the monogenity of totally complex pure octic fields

TL;DR

This paper investigates whether totally complex pure octic fields generated by roots of negative integers admit generators of power integral bases beyond the known root, providing an efficient computational method and extensive results for fields with specific parameters.

Contribution

The paper introduces an efficient method to find all generators of power integral bases in totally complex pure octic fields with large coefficients, extending previous knowledge.

Findings

Identified 2024 such fields with coefficients between -5000 and 0.

Found that the known generator is often the only one in these fields.

Developed a computational approach for large coefficient bounds.

Abstract

Let $0,1\ne m\in Z$ and $\alpha=\sqrt[8]{m}$. According to the results of I. Ga\'al and L. El Fadil, $\alpha$ generates a power integral basis in $K=Q(\alpha)$, if and only if $m$ is square-free and $m\not\equiv 1\;(\bmod\; 4)$. In the present paper we consider totally complex pure octic fields, that is the case $m<0$, with $m$ satisfiying the above property. In this case $(1,\alpha,\alpha^2,\ldots,\alpha^7)$ is an integral basis. Our purpose is to investigate whether $K$ admits any other generators of power integral bases, inequivalent to $\alpha$. We present an efficient method to calculate generators of power integral bases in this type of fields with coefficients $<10^{200}$ in the above integral basis. We report on the results of our calculation for this type of fields with $0>m>-5000$, which yields 2024 fields.