As mentioned by @MichaelZieve in his comment re Quadratic residue, Chebotarev's density theorem was preceded by an allegedly much easier theorem of Frobenius (Mike Zieve is certainly not the only one to mention that the Frobenius theorem is much easier than Chebotarev) -- the difference between the two theorems is that Frobenius talks of conjugacy in $S_n,$ while Chebotarev of conjugacy in the Galois group itself. Does anyone know what the "easier" argument is? Is there a good reference?
You can find Frobenius's theorem (and proof) on p.134 of Janusz, Algebraic Number Theory, 1973, and many other places. It really is easier. It was known long before Chebotarev's theorem, and Chebotarev had to come up with new ideas to prove his theorem (which in turn helped Artin prove his reciprocity law).