Faithful Artin induction and the Chebotarev density theorem

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.