Dedekind zeta function

For a number field $K$, the Dedekind zeta function is formally

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 $\det (1-T\rho (\mathrm{Frob}))=1-T^f$. 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.

Unique factorization of nonzero ideals and multiplicativity of the ideal norm give the formal identitywhere the product ranges over nonzero prime ideals.

For a quadratic field $K$ of discriminant $D$,where ${\chi }_D(p)$ is $1$, $-1$, or $0$ according as $(p)$ splits, remains prime, or ramifies in ${\mathcal{O}}_K$. Comparing the three possible local Euler factors proves the formal identity.

For a number field of signature $(r_1,r_2)$, roots-of-unity count $w_K$, regulator $R_K$, class number $h_K$, and discriminant $d_K$,

For an imaginary quadratic field of fundamental discriminant $D<-4$,Equivalent finite character-sum forms follow by evaluating the odd quadratic Dirichlet L-function at one.