On certain pure sextic fields related to a problem of Hasse. (English) Zbl 1404.11124
Summary: An algebraic number ring is monogenic, or one-generated, if it has the form \(\mathbf{Z}[\alpha]\) for a single algebraic integer \(\alpha\). It is a problem of Hasse to characterize, whether an algebraic number ring is monogenic or not. In this note, we prove that if \(m\) is a square-free rational integer, \(m\equiv 1\pmod 4\) and \(m\not\equiv\pm 1\pmod 9\), then the pure sextic field \(L=\mathbb{Q}(\root6\of {m})\) is not monogenic. Our results are illustrated by examples.
References:
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