$α$-monogeneity of pure number fields: criterion and density
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915715012886528 |
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| author | Nguyen-Dang, Khai-Hoan Hung, Nguyen Thai |
| author_facet | Nguyen-Dang, Khai-Hoan Hung, Nguyen Thai |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20232 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $α$-monogeneity of pure number fields: criterion and density Nguyen-Dang, Khai-Hoan Hung, Nguyen Thai Number Theory 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. |
| title | $α$-monogeneity of pure number fields: criterion and density |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.20232 |