Restricting the Splitting Types of a Positive Density Set of Places in Number Field Extensions

Abstract

We prove necessary and sufficient conditions for a finite group $G$ with an ordering of $G$-extensions to satisfy the following property: for every positive density set of places $A$ of a number field $K$ and every splitting type given by a conjugacy class $c$ in $G$, $0\%$ of $G$-extensions avoid this splitting type for each $p\in A$.