Faithful Artin induction and the Chebotarev density theorem

TL;DR

This paper proves a new result about faithful irreducible characters of finite groups and applies it to establish an effective Chebotarev density theorem, leading to insights on class groups in number field families.

Contribution

It introduces a novel approach to generating faithful characters via monomial characters and applies this to derive an effective Chebotarev density theorem for G-extensions.

Findings

Almost all G-extensions satisfy the effective Chebotarev density theorem.

New relationships between faithful characters and monomial characters.

Implications for class group behavior in number field families.

Abstract

Given a finite group G, we prove that the vector space spanned by the faithful irreducible characters of G is generated by the monomial characters in the vector space. As a consequence, we show that in any family of G-extensions of a fixed number field F, almost all are subject to a strong effective version of the Chebotarev density theorem. We use this version of the Chebotarev density theorem to deduce several consequences for class groups in families of number fields.