The average genus number for pure fields of prime degree

Abstract

Let $\ell \ge 5$ 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.

10 pages