$α$-monogeneity of pure number fields: criterion and density

TL;DR

The paper provides a complete local–global criterion for when a pure number field $K=\mathbb{Q}(\alpha)$ with $\alpha^n=m$ is $\alpha$-monogenic, proving $\mathcal{O}_K=\mathbb{Z}[\alpha]$ exactly when $m$ is square-free and $\nu_p(m^p-m)=1$ for all $p\mid n$ via Dedekind’s index theorem. Building on this criterion, it derives a precise natural density for $m$ yielding $\alpha$-monogenic fields: $\delta_n=\frac{6}{\pi^2}\prod_{p\mid n}\frac{p}{p+1}$, obtained by independence of local obstructions across primes and refined by arithmetic progressions with explicit local factors. The work also analyzes local exclusions $\mathcal{E}_p=\{m: m^p\equiv m \pmod{p^2}\}$ and gives discriminant-order asymptotics for counting $\mathbb{Q}(\sqrt[n]{m})$ with maximal orders, including explicit constants for small $n$. Overall, the results provide a sharp, thin-family density picture for monogenic pure fields and complement broader coefficient-space monogeneity results. The methods combine a concise Dedekind-index argument with a probabilistic, independence-based density framework, yielding exact densities and accessible local–global criteria relevant to algebraic number theory and arithmetic statistics.

Abstract

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

Theorems & Definitions (41)

• Definition 2.1
• Definition 2.2
• Theorem 2.3: Dedekind’s index theorem
• Theorem 2.4
• Lemma 2.5
• proof
• Theorem 2.6
• proof
• Remark 2.7: On the reduction from $p^r$ to $p$
• Corollary 2.8