Global Class Field Theory Caltech

2.1. The Cebotarev density theorem. We have seen in the proof of Theorem 1.3 that for any nontrivial ray class character χ : H → C× we could show nonvanishing L(1, χ) = 0 provided that χ factors through J/NL/KJ L · P for some finite extension L/K. By Theorem 1.2 the subgroup P is the norm subgroup of the ray class field K(m)/K. Hence we get nonvanishing for all nontrivial χ and this allows to prove the following strengthening of the the Frobenius density theorem (the analogue of Dirichlet’s theorem on primes in arithmetic progressions for the number field K).