Counting metacyclic fields

Abstract

By a metacyclic field $K$ we mean the Galois closure of a pure field $\mathbb{Q}(\sqrt[\ell]{D})$, $D\in\mathbb{Z}$, of odd prime degree $\ell$. Let $\mathcal{M}_\ell$ denote the collection of isomorphism classes of metacyclic fields of degree $\ell(\ell-1)$. Write $N_\ell(X)=\#\{K\in\mathcal{M}_\ell: |\Delta_K|\leq X\}$ for the associated counting function, where $\Delta_K$ denotes the discriminant of $K$. We show $$N_\ell(X)\sim A_\ell X^{\frac{1}{(\ell-1)^2}}(\log X)^{\ell-2}\,,$$ for an explicit constant $A_\ell$. We express $A_\ell$ as a rational number times a product over primes of a degree $\ell$ polynomial in $1/p$.