$\alpha$-monogeneity of pure number fields: criterion and density

TL;DR

This paper provides a criterion for $ ext{alpha}$-monogeneity in pure number fields and calculates the natural density of such fields, with implications for their arithmetic properties and distribution.

Contribution

It offers a new proof of the $ ext{alpha}$-monogeneity criterion and derives explicit density formulas and asymptotic behaviors for pure number fields.

Findings

Criterion: $ ext{Z}[ ext{alpha}]= ext{O}_K$ iff $m$ is square-free and $ u_p(m^p-m)=1$ for all $p|n$.

Explicit density $rac{6}{ ext{pi}^2} imes ext{product over } p|n$.

Density independence across primes and asymptotic discriminant estimates.

Abstract

For pure extensions $K=\mathbb{Q}(\alpha)$ with $\alpha^n=m$, we give a short proof, based only on Dedekind's index theorem, of the $\alpha$-monogeneity criterion: $\mathbb{Z}[\alpha]=\mathcal{O}_K$ if and only if $m$ is square-free and $\nu_p(m^p-m)=1$ for every prime $p\mid n$. We then derive an explicit natural density $\delta_n=\frac{6}{\pi^2}\prod_{p\mid n}\frac{p}{p+1}$, independence across primes, refinements in arithmetic progressions, and discriminant-order asymptotics.