On monogenity of certain pure number fields defined by $x^{2^u\cdot 3^v\cdot 5^t}-m$

Abstract

Let $K=\mathbb{Q}(\alpha )$ be a pure number field generated by a root $\alpha$ of a monic irreducible polynomial $F(x)=x^{2^u\cdot 3^v\cdot 5^t}-m$, with $m\ne \pm 1$ a square free rational integer, $u$, $v$ and $t$ three positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md4$, $m\not\equiv \pm 1\md9$, and $m\not\in\{\pm 1, \pm 7\}\md{25}$, then $K$ is monogenic. But if {$m\equiv 1\md{4}$} or $m\equiv 1\md9$ or $m\equiv -1\md9$ and $u=2k$ for some odd integer $k$ or $u\ge 2$ and $m\equiv 1\md{25}$ or $m\equiv -1\md{25}$ and $u=2k$ for some odd integer $k$ or $u=v=1$ and $m\equiv \pm 82\md{5^4}$, then $K$ is not monogenic.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2106.01252, arXiv:2112.01133, arXiv:2111.05899, arXiv:2106.00004; text overlap with arXiv:2203.13353