标题： 纯次数为素数的域的平均亏格数 显示英文标题

标题： The average genus number for pure fields of prime degree

摘要： 设$\ell\geq 5$为素数。 设$\mathcal{F}_\ell$为次数为$\ell$的纯数域$\mathbb{Q}(\sqrt[\ell]{a})$的同构类的集合，按其判别式的绝对值进行排序。 2018年， Benli 对$\mathcal{F}_\ell$证明了一个计数定理，推广了Cohen和Morra在$\ell=3$时之前的定理。 我们证明，次数为$\ell$的纯域中，类数为一的比例渐近于$(A_\ell \log X)^{-1}$，并且次数为$\ell$的纯域的平均类数渐近于$B_\ell(\log X)^{\ell-1}$。 两者$A_\ell$和$B_\ell$都明确地表示为素数的乘积。 显示英文摘要

摘要： 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.