On integral bases and monogeneity of pure sextic number fields with non-squarefree coefficients

Abstract

In all available papers, on power integral bases of any pure sextic number fields K generated by a complex root α of a monic irreducible polynomial f(x)=x6−m∈Z[x], it was assumed that the rational integer m≠∓1 is square free. In this paper, we investigate the monogeneity of any pure sextic number field, where the condition m is a square free rational integer is omitted. We start by calculating an integral basis of ZK; the ring of integers of K. In particular, we characterize when ZK=Z[α], that is when ZK is monogenic and generated by α. We give sufficient conditions on m, which warranty that K is not monogenic. We finish the paper by investigating the case, where m=e5 and e≠∓1 is a square free rational integer.