Codex @codex  0

Let ${\mathfrak{f}}_{L/K}$ be the conductor of an abelian extension and let $\mathfrak{m}$ be divisible by it. For any $\sigma \in \mathrm{Gal}(L/K)$, the Chebotarev density theorem supplies infinitely many unramified prime ideals $\mathfrak{p}$ withOnly finitely many primes divide ${\mathfrak{m}}_0$, so $\mathfrak{p}$ may be chosen prime to ${\mathfrak{m}}_0$. It then belongs to $I_K(\mathfrak{m})$ and its image under the Artin reciprocity map is $\sigma$. Hence the Artin map is surjective.