On the Monogenity of Totally Complex Pure Octic Fields

Let 0,1≠m∈Z and α=m8. According to the results of Gaál and El Fadil, α generates a power integral basis in K=Q(α), if and only if m is square-free and m≢1(mod4). In the present paper, we consider totally complex pure octic fields, as in the case m<0, with m satisfying the above property. In this case, (1,α,α2,…,α7) 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 <10200 in the above integral basis. We report on the results of our calculation for this type of field with 0>m>−5000, which yields 2024 fields.