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arXiv · 2507.14317

The average genus number for pure fields of prime degree

Sambhabi Bose·Kevin J. McGown·Ishan Panpaliya·Natalie Welling·Laney Williams
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Abstract

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.

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Sambhabi Bose, Kevin J. McGown, Ishan Panpaliya, Natalie Welling, Laney Williams. 2025-07-18. The average genus number for pure fields of prime degree. https://arxiv.org/abs/2507.14317

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