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I'm wondering if there is a proof of the Chebotarev density theorem that does not require the use of any big results in class field theory, such as Artin Reciprocity.

As I understand it, the main difficulty comes in establishing the abelian case, in which the idea of proof is similar to Dirichlet's theorem. Then we run into the issue of the non-vanishing of the L-function, which is proven by showing that their product is the Dedekind zeta function. This last statement requires Artin reciprocity.

I would like to know if there's another way of getting around this or if the last statement doesn't actually require any advanced class field theory.

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  • $\begingroup$ Looks like math.leidenuniv.nl/~hwl/PUBLICATIONS/1996d/art.pdf has an outline of such a proof at the end. I may update this with an answer if/when I work through it. $\endgroup$
    – Cyclicduck
    Apr 20, 2019 at 4:21
  • $\begingroup$ Did you try with an example ? Say $L(s,\rho)$ with $\rho$ the dimension $2$ representation of $Gal(L/Q)$ and $K$ the splitting field of $x^3+x+1$. Class field theory or CM elliptic curves says $L(s,\rho) = L(s,f)$ with $f $ a modular form $\in S_1(\Gamma_0(D),(\frac{.}{D})),D=Disc(x^3+x+1)$ $\endgroup$
    – reuns
    Apr 20, 2019 at 5:55
  • $\begingroup$ I don't know any CFT or CM (or even what that L-function is with $\rho$ 2d) $\endgroup$
    – Cyclicduck
    Apr 21, 2019 at 4:05
  • $\begingroup$ $Gal(L/Q) = S_3$ then $L(s,\rho) = \prod_p \exp(\sum_k\frac{p^{-sk}}{k} Tr(\rho(Frob_p^k)))$ where $Tr(\rho) = (2,-1,-1,0,0,0)$ is the character of the dihedral representation. Then $L(s,\rho) = L(s,\phi)$ for some abelian character of $Gal(L/F), F= Q(\sqrt{D})$ and CFT says $L(s,\phi)=L(s,\psi)$ with $\psi$ a Hecke character with a simple expression in term of congruences and ideal class $\endgroup$
    – reuns
    Apr 21, 2019 at 16:55

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How about this?

The abstract says "We give two proofs of the Chebotarev density theorem: Chebotarev’s original proof [Tsc26], which predates class field theory, and a more recent proof due to Lagarias and Odlyzko [LO77] and refined by Serre [Ser81] which gives an “effective” error term. ".

Although avoiding Artin reciprocity law, it seems to appear similar theorem for cyclotomic extension(Lemma 5.4,5.5).

It seems to prove for in the order of cyclotomic extension case, finite Abelian extension case, finite Galois extension case.

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