On power integral bases of certain pure number fields defined by $x^{3^r\cdot 7^s}-m$
Volume 169 / 2022
Colloquium Mathematicum 169 (2022), 307-317
MSC: Primary 11R04, 11Y40; Secondary 11R21.
DOI: 10.4064/cm8574-6-2021
Published online: 11 March 2022
Abstract
Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{3^r\cdot 7^s}-m\in \mathbb {Z}[x]$ with $m\neq \pm 1$ a square free integer, and $r$ and $s$ two positive integers. In this paper, we study the monogeneity of $K$. The case $r=0$ or $s=0$ has been studied in [H. Ben Yakkou and L. El Fadil, Int. J. Number Theory 17 (2021)]. We prove that if $\def\md#1{\ \mbox{(mod }{#1})} m\not \equiv \pm 1\md 9$ and $\overline {m}\not \in \{\mp 1,\mp 18,\mp 19\} \md {49}$, then $K$ is monogenic. But if $r\ge 3$ and $\nu _3(m^2-1)\ge 4$ or $s\ge 3$, $\def\md#1{\ \mbox{(mod }{#1})}{m}\equiv \pm 1 \md {7}$, and $\nu _7(m^6-1)\ge 3$, then $K$ is not monogenic. Some illustrating examples are given.
