On power integral bases for certain pure number fields defined by $x^{2\cdot 3^k}-m$
Volume 201 / 2021
Acta Arithmetica 201 (2021), 269-280
MSC: Primary 11R04, 11R21; Secondary 11Y40.
DOI: 10.4064/aa210402-8-5
Published online: 4 November 2021
Abstract
Let $K=\mathbb Q (\alpha )$ be the number field generated by a complex root $\alpha $ of a monic irreducible polynomial $f(x)=x^{2\cdot 3^k}-m$ with {$m\neq \pm 1$} a square free rational integer and $k$ a positive integer. We prove that if $m \equiv 2 \mbox { or } 3\def\md#1{\ \mbox{(mod }{#1})}\md 4$ and {$m\not \equiv \pm 1\def\md#1{\ \mbox{(mod }{#1})}\md 9$}, then the field $K$ is monogenic, while if $m \equiv 1\def\md#1{\ \mbox{(mod }{#1})}\md 4$ or $m\equiv 1\def\md#1{\ \mbox{(mod }{#1})}\md 9$ or $k\ge 3$ and $m\equiv -1\def\md#1{\ \mbox{(mod }{#1})}\md {81}$, then $K$ is not monogenic.
