On certain pure sextic fields related to a problem of Hasse

Abstract

An algebraic number ring is monogenic, or one-generated, if it has the form [Formula: see text] for a single algebraic integer [Formula: see text]. It is a problem of Hasse to characterize, whether an algebraic number ring is monogenic or not. In this note, we prove that if [Formula: see text] is a square-free rational integer, [Formula: see text] and [Formula: see text], then the pure sextic field [Formula: see text] is not monogenic. Our results are illustrated by examples.