The average genus number for pure fields of prime degree

TL;DR

This paper studies the distribution of genus numbers in pure number fields of prime degree, providing asymptotic formulas for the proportion with genus number one and the average genus number, expressed explicitly in terms of prime products.

Contribution

It establishes explicit asymptotic formulas for the proportion and average of genus numbers in pure fields of prime degree, extending previous counting results.

Findings

Proportion of pure fields with genus number one is asymptotic to (A_ell log X)^{-1}.

Average genus number for pure fields is asymptotic to B_ell (log X)^{ell-1}.

Both constants are explicitly given as prime products.

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.