On monogenity of certain pure number fields defined by $x^{p^r}-m$

TL;DR

This paper investigates the conditions under which certain pure number fields defined by polynomials of the form x^{p^r} - m are monogenic, providing criteria based on p-adic valuations of m and m^p - m.

Contribution

It establishes new criteria for the monogenity of pure number fields generated by x^{p^r} - m, depending on p-adic valuations, extending previous understanding.

Findings

If ν_p(m^p - m) = 1, then the field is monogenic.

If r ≥ p and ν_p(m^p - m) > p, then the field is not monogenic.

Examples illustrate the application of the criteria.

Abstract

Let $K = \mathbb{Q} (\alpha) $ be a pure number field generated by a complex root $\alpha$ a monic irreducible polynomial $ F(x) = x^{p^r} -m$, with $ m \neq 1 $ is a square free rational integer, $p$ is a rational prime integer, and $r$ is a positive integer. In this paper, we study the monogenity of $K$. We prove that if {{$\nu_p(m^p-m)=1$}}, then $K$ is monogenic. But if $r\ge p$ and {$\nu_p(m^{p}-m)> p$}, then $K$ is not monogenic. Some illustrating examples are given.