In ch. VII, §13, pp. 545-546 of the English translation of Algebraic Number Theory by J. Neukirch, one finds a 1-page proof of the Chebotarev density theorem. This proof seems fairly harmless, and seems to only assume the following two results:
(1) The "class field theoretic isomorphism" between ray class groups and corresponding Abelian extensions (which of course is not itself a trivial result).
(2) What he calls the "generalised Dirichlet density theorem", which he proves on the previous page.
Now I was always under the impression that the Chebotarev density theorem, being what one might call a non-Abelian generalisation of the "generalised Dirichlet density theorem", would essentially depend upon the theory of the (non-Abelian) Artin $L$-series.
But the only place where the Artin $L$-series seem to feature in the proof, is when he proves the non-vanishing of the non-principal generalised Dirichlet $L$-series at $s=1$. The use of Artin $L$-series here is of course purely a matter of convenience, as there are more elementary ways of demonstrating the non-vanishing at $s=1$.
So my question is: Does the proof of the Chebotarev density theorem essentially depend upon the introduction of Artin $L$-series, or can one prove it without them if one demonstrates the non-vanishing of the generalised Dirichlet $L$-series at $s=1$ by other means? Is there some Artinian subtlety that I am missing?
Thank you for your attention.
Best Answer
The point is that while the Chebotarev density theorem is about how often some element $\sigma$ of a possibly non-abelian Galois group of number fields (or more generally global fields) arises as a Frobenius element at some (unramified) prime ideal, by a trick of Deuring the proof can be reduced to the case of the Galois extension corresponding to the subgroup $\langle \sigma\rangle$ generated by $\sigma$, i.e., to a cyclic Galois extension. In this process, the base field changes to the fixed field of $\sigma$, so you need to formulate the Chebotarev density theorem over fairly general number fields in order to use this trick. And once you are in the setting of cyclic Galois extensions, class field theory lets you focus on $L$-functions of finite-order Hecke characters.
Historically, Chebotarev proved his density theorem before class field theory and without Deuring's trick. In fact he proved his density theorem only for Galois extensions of $\mathbf Q$, so he could not take advantage of a reduction to the cyclic case by increasing his base field. The discussion here has references to the original papers by Chebotarev.