On monogenity of certain pure number fields defined by \(x^{{2}^{u}.3^{v}} - m\)

Abstract

Let \(K = \mathbb {Q} (\alpha)\) be a pure number field generated by a complex root \(\alpha \) of a monic irreducible polynomial \(F(x) = x^{{2}^{u}.3^{v}} - m\), with \(m \neq \pm 1 \) a square free rational integer, u, and v two positive integers. In this paper, we study the monogenity of K. The cases \(u = 0\) and \(v=0\) have been previously studied by the first author and Ben Yakkou. We prove that if m ≢ 1 (mod 4) and m ≢ \(\pm\)1 (mod 9), then K is monogenic. But if \(m \equiv 1\) (mod 4) or \(m \equiv 1 \) (mod 9) or \(u = 2\) and \(m \equiv -1\) (mod 9), then K is not monogenic. Some illustrating examples are given too.