Document Type : Original Article
Authors
1
Department of Mathematics, Polydisciplinary Faculty of Taza, Sidi Mohamed Ben Abdellah University, Taza, Morocco
2
Department of Mathematics, Faculty of Sciences, Moulay Ismail University, Meknes, Morocco
3
Department of mathematics, Polydisciplinary Faculty of Taza, Sidi Mohamed Ben Abdellah University, Taza, Morocco
Abstract
Let $K = \mathbb{Q} (\alpha)$ be a pure number field generated by $\alpha$ a root of a monic irreducible polynomial $ F(x)\, = \, x^{3^r\cdot 5^s \cdot 7^t}-m \in \mathbb{Z}[x]$, where $m \neq \pm 1$ is a square-free rational integer, $r$, $s$, and $t$ are three positive rational integers. The aim of this paper is to study the problem of monogeneity of the field $K$. More precisely, we provide explicit conditions on $r, s, t,$ and $m$ for which $K$ is monogenic. We show that if $ m \not \equiv \pm 1\,( mod \, 9)$, $\overline{m} \not \in \{ \overline{\pm 1},\overline{\pm7}\} \,(mod \, {25}) $, and $ \overline{m} \not \in \{ \overline{\pm1}, \overline{\pm18}, \overline{\pm19}\} \, ( mod \, {49}) $, then $K$ is monogenic. In addition, we prove the existence of infinite families of nonmonogenic number fields of degree $n=3^r\cdot 5^s \cdot 7^t$. At the conclusion of this work, a few illustrative examples are provided.
Keywords
Subjects
