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

Let $K = \mathbb{Q} (α)$ be a pure number field generated by a root $α$ of a monic irreducible polynomial $F(x) = x^{2^u\cdot 3^v\cdot 5^t}-m$, with $m \neq \pm 1$ 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.

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