= Dedekind zeta function as a permutation Artin L-function
{title2=$\zeta_K(s)=L_{\mathbb Q}(\mathbb C[\operatorname{Hom}(K,E)],s)$}
Let $E$ be a finite normal closure of the <number field> $K$. The <permutation representation> of its <Galois group> on the embeddings of $K$ into $E$ has one trivial constituent, because the action is transitive. At a prime, the <inertia group> invariant basis consists of sums over inertia orbits of embeddings; <Frobenius automorphism> permutes them in cycles of lengths equal to the <residue degrees> of the primes of $K$. A cycle of length $f$ contributes $\det(1-T\rho(\operatorname{Frob}))=1-T^f$. Hence the local factors equal those of the <Dedekind zeta function>, including at ramified primes. If nontrivial irreducible <Artin L-functions> are holomorphic and nonzero at one, this factorization and the simple pole of the <Riemann zeta function> show that $\zeta_K$ has a simple pole there.
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