The Distribution of Number Fields with Wreath Products as Galois Groups

Abstract

Let $G$ be a wreath product of the form $C_2 \wr H$, where $C_2$ is the cyclic group of order 2. Under mild conditions for $H$ we determine the asymptotic behavior of the counting functions for number fields $K/k$ with Galois group $G$ and bounded discriminant. Those counting functions grow linearly with the norm of the discriminant and this result coincides with a conjecture of Malle. Up to a constant factor these groups have the same asymptotic behavior as the conjectured one for symmetric groups.