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

TL;DR

This paper establishes precise conditions under which a positive density set of places in a number field ensures that all G-extensions exhibit a specific splitting type, linking group theory and number field properties.

Contribution

It provides necessary and sufficient criteria connecting the structure of G-extensions with the distribution of splitting types across places in number fields.

Findings

Characterization of G-extensions with prescribed splitting behavior

Conditions ensuring all G-extensions realize a given splitting type

Link between group structure and distribution of splitting types

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 $p\in A$.