The average genus number for pure fields of prime degree
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916851506741248 |
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| author | Bose, Sambhabi McGown, Kevin J. Panpaliya, Ishan Welling, Natalie Williams, Laney |
| author_facet | Bose, Sambhabi McGown, Kevin J. Panpaliya, Ishan Welling, Natalie Williams, Laney |
| contents | Let $\ell\geq 5$ be prime. Let $\mathcal{F}_\ell$ be the collection of (isomorphism classes of) pure number fields $\mathbb{Q}(\sqrt[\ell]{a})$ of degree $\ell$, ordered by the absolute value of their discriminant. In 2018, Benli proved a counting theorem for $\mathcal{F}_\ell$, generalizing a previous theorem of Cohen and Morra when $\ell=3$. We prove that the proportion of pure fields of degree $\ell$ with genus number one is asymptotic to $(A_\ell \log X)^{-1}$ and that the average genus number for pure fields of degree $\ell$ is asymptotic to $B_\ell(\log X)^{\ell-1}$. Both $A_\ell$ and $B_\ell$ are expressed explicitly as a product over primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14317 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The average genus number for pure fields of prime degree Bose, Sambhabi McGown, Kevin J. Panpaliya, Ishan Welling, Natalie Williams, Laney Number Theory Primary: 11R45, 11N45, Secondary: 11R29 Let $\ell\geq 5$ be prime. Let $\mathcal{F}_\ell$ be the collection of (isomorphism classes of) pure number fields $\mathbb{Q}(\sqrt[\ell]{a})$ of degree $\ell$, ordered by the absolute value of their discriminant. In 2018, Benli proved a counting theorem for $\mathcal{F}_\ell$, generalizing a previous theorem of Cohen and Morra when $\ell=3$. We prove that the proportion of pure fields of degree $\ell$ with genus number one is asymptotic to $(A_\ell \log X)^{-1}$ and that the average genus number for pure fields of degree $\ell$ is asymptotic to $B_\ell(\log X)^{\ell-1}$. Both $A_\ell$ and $B_\ell$ are expressed explicitly as a product over primes. |
| title | The average genus number for pure fields of prime degree |
| topic | Number Theory Primary: 11R45, 11N45, Secondary: 11R29 |
| url | https://arxiv.org/abs/2507.14317 |