Chebotarev density theorem

For a finite Galois extension $L/K$ and a conjugacy class $C$ in $G=\mathrm{Gal}(L/K)$, the unramified primes with Frobenius conjugacy class $C$ have Dirichlet density $|C|/|G|$. In particular,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.

The logarithm from the Euler product of an Artin L-function has first-prime-power term $\sum \chi (g_{\mathfrak{p}})(N\mathfrak{p}{)}^{-s}$. Higher powers are uniformly bounded as real $s\downarrow 1$: $|\chi (g)|\le \chi (1)$ and ${\sum }_{\mathfrak{p}}{\sum }_{m\ge 2}(N\mathfrak{p}{)}^{-ms}/m\le 2{\sum }_{\mathfrak{p}}(N\mathfrak{p}{)}^{-2}<\infty$. The finitely many ramified local factors also have bounded logarithms. The Artin L-function order at one therefore proves the displayed estimate. Expanding a conjugacy-class indicator by character orthogonality leaves only the trivial character's logarithmic term, with coefficient $|C|/|G|$. This proves Dirichlet density directly; a natural-density assertion needs additional analytic arguments.