Revista de la
Unión Matemática Argentina
On power integral bases of certain pure number fields defined by $x^{84}-m$
Lhoussain El Fadil and Mohamed Faris
Volume 65, no. 1 (2023), pp. 197–211    

https://doi.org/10.33044/revuma.2836

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Abstract

Let $K=\mathbb{Q}(\alpha)$ be a pure number field generated by a complex root $\alpha$ of a monic irreducible polynomial $F(x) = x^{84}-m$, with $m\neq\pm 1$ a square-free integer. In this paper, we study the monogeneity of $K$. We prove that if $m \not\equiv 1\pmod{4}$, $m \not\equiv \pm 1\pmod{9}$, and $\overline{m} \not\in \{\pm \overline{1},\pm \overline{18}, \pm \overline{19}\} \pmod{49}$, then $K$ is monogenic. But if $m \equiv 1\pmod{4}$ or $m \equiv\pm 1\pmod{9}$ or $m \equiv 1 \pmod{49}$, then $K$ is not monogenic. Some illustrating examples are given.

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