= Chebotarev density theorem
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For a finite <Galois extension> $L/K$ and a <conjugacy class> $C$ in $G=\operatorname{Gal}(L/K)$, the <unramified> primes with <Frobenius conjugacy class> $C$ have <Dirichlet density> $|C|/|G|$. In particular,
$$
\sum_{\mathfrak p:\,\operatorname{Frob}_{\mathfrak p}\in C}(N\mathfrak p)^{-s}
=\frac{|C|}{|G|}\log\frac1{s-1}+O(1)\quad(s\downarrow1).
$$
One proof uses <Brauer induction>, the <Artin L-function order at one>, and <character orthogonality>. The trivial <Artin L-function> is the <Dedekind zeta function> of $K$. The full theorem also gives the natural density, but this logarithmic estimate alone establishes the <Dirichlet density> assertion.
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