Abstract
Let \(K\) be a pure number field generated by a root of a monic irreducible polynomial \(F(x)=x^{2^r\cdot5^s\cdot 7^t}-m\in \mathbb{Z}[x]\), where \(m\neq \pm 1\) is a square free integer, \(r\), \(s\), and \(t\) are three positive integers. In this paper, we study the monogenity of \(K\). We prove that if \(m\not\equiv 1 (\text{mod }{4})\), \(\overline{m}\not\in\{\pm\overline{1},\pm \overline{7}\} (\text{mod }{25})\), and \(\overline{m}\not\in\{\pm \overline{1},\pm \overline{18},\pm \overline{19}\} (\text{mod }{49})\), then \(K\) is monogenic. But if \(r\geq2\) and \(m\equiv 1 (\text{mod }{16})\) or \(r=1\), \(s\geq2\), and \(m\equiv \pm1 (\text{mod }{125})\) or \(r\geq2\) and \(m\equiv 1 (\text{mod }{25})\) or \(t\geq 3\), \(\nu_7(m^6-1)\geq 4\), and \(\overline{m}\in\{\overline{1},\overline{18},-\overline{19}\} (\text{mod }{49})\), then \(K\) is not monogenic. Some illustrating examples will be given.
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The authors are very grateful to the anonymous referee whose valuable comments and suggestions have tremendously improved the quality of this paper.
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Kchit, O., Choulli, H. On Monogenity of Certain Pure Number Fields Defined by \(x^{2^r\cdot5^s\cdot 7^t}-m\). P-Adic Num Ultrametr Anal Appl 14 (Suppl 1), S1–S9 (2022). https://doi.org/10.1134/S2070046622050016
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DOI: https://doi.org/10.1134/S2070046622050016