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

TL;DR

This paper investigates the conditions under which certain pure number fields generated by roots of specific polynomials are monogenic, providing criteria based on congruences of the defining integer m.

Contribution

It establishes new sufficient and necessary conditions for the monogenity of pure number fields defined by polynomials of the form x^{2^u 3^v 5^t} - m, expanding understanding of their algebraic structure.

Findings

If m does not satisfy certain congruences, K is monogenic.

If m satisfies specific congruences, K is not monogenic.

Provides explicit criteria based on modular conditions for monogenity.

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 \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.