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 $N_{\ell }(X)=\#\{K\in {\mathcal{M}}_{\ell }:|{\Delta }_K|\le X\}$ 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 $1/p$.