On the monogenity of totally complex pure octic fields

Abstract

Let $0,1\ne m\in Z$ and $α=\sqrt[8]{m}$. According to the results of I. Gaál and L. El Fadil, $α$ generates a power integral basis in $K=Q(α)$, 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 $(1,α,α^2,\ldots,α^7)$ is an integral basis. Our purpose is to investigate whether $K$ admits any other generators of power integral bases, inequivalent to $α$. 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.