On monogenity of certain pure number fields defined by $x^{2^r\cdot7^s}-m$

IF 0.4 Q4 MATHEMATICS Boletim Sociedade Paranaense de Matematica Pub Date : 2022-12-28 DOI:10.5269/bspm.62352
L. El Fadil, O. Kchit
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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.
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关于由x^{2^r\cdot7^s}-m$定义的纯数域的单性
设$K$是单不可约多项式$F(x)=x^{2^r\cdot7^s}-m\in\mathbb{Z}[x]$的复根生成的纯数域,其中$m\neq\pm1$是一个无平方整数,$r$和$s$是两个正整数。本文研究了$K$的单胚性。我们证明了如果$m\not \equiv 1\md{4}$和$\overline{m}\not \in \{\pm\overline}1},\pm\overline{18},\ pm \overline{19}\}\md{49}$,那么$K$是单基因的。但是,如果$r\geq2$和$m\equiv 1\md{16}$或$s\geq3$、$\overline{m}\in \{\overline}1}、\overline{18}、-\overline{19}\}\md{49}$和$\nu_7(m^6-1)\geq4$,则$K$不是单基因的。文末给出了一些实例。
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来源期刊
CiteScore
1.40
自引率
0.00%
发文量
140
审稿时长
25 weeks
期刊最新文献
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