{"title":"On monogenity of certain pure number fields defined by $x^{2^r\\cdot7^s}-m$","authors":"L. El Fadil, O. Kchit","doi":"10.5269/bspm.62352","DOIUrl":null,"url":null,"abstract":"Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{2^r\\cdot7^s}-m\\in \\mathbb{Z}[x]$, where $m\\neq \\pm 1$ is a square free integer, $r$ and $s$ are two positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\\not\\equiv 1\\md{4}$ and $\\overline{m}\\not\\in\\{\\pm \\overline{1},\\pm \\overline{18},\\pm \\overline{19}\\} \\md{49}$, then $K$ is monogenic. But if $r\\geq 2$ and $m\\equiv 1\\md{16}$ or $s\\geq 3$, $\\overline{m}\\in\\{ \\overline{1}, \\overline{18}, -\\overline{19}\\} \\md{49}$, and $\\nu_7(m^6-1)\\geq 4$, then $K$ is not monogenic. Some illustrating examples are given at the end of the paper.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Boletim Sociedade Paranaense de Matematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5269/bspm.62352","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{2^r\cdot7^s}-m\in \mathbb{Z}[x]$, where $m\neq \pm 1$ is a square free integer, $r$ and $s$ are two positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md{4}$ and $\overline{m}\not\in\{\pm \overline{1},\pm \overline{18},\pm \overline{19}\} \md{49}$, then $K$ is monogenic. But if $r\geq 2$ and $m\equiv 1\md{16}$ or $s\geq 3$, $\overline{m}\in\{ \overline{1}, \overline{18}, -\overline{19}\} \md{49}$, and $\nu_7(m^6-1)\geq 4$, then $K$ is not monogenic. Some illustrating examples are given at the end of the paper.