On power integral bases of certain pure number fields defined by $x^{2^u\cdot 3^v}-m$

Abstract

Let $K=\mathbb{Q}(\alpha )$ be a pure number field generated by a complex root $\alpha$ of a monic irreducible polynomial $F(x)=x^{2^u\cdot 3^v}-m$, with $m\ne \pm 1$ a square free rational integer, $u$, and $v$ two positive integers. In this paper, we study the monogenity of $K$. The case $u=0$ or $v=0$ has been previously studied. We prove that if $m\not\equiv 1$ (mod4) and $m\not\equiv \pm 1$ (mod9), then $K$ is monogenic. But if $m\equiv 1$ (mod4) or $m\equiv 1$ (mod9), then $K$ is not monogenic. Some illustrating examples are given too.

Submitted. arXiv admin note: substantial text overlap with arXiv:2102.01967, arXiv:2106.00004, arXiv:2006.11230

References in corpus (2)

• Integral bases and monogenity of pure fields
• Binomial Thue equations and power integral bases in pure quartic fields