Factorization of the Dedekind zeta function of an abelian number field

The Dedekind zeta function of an abelian number field factors as a product of Dirichlet L-functions as follows. Let $K$ be an abelian number field, i.e. $K/\mathbb{Q}$ is Galois and $\mathrm{Gal}(K/\mathbb{Q})$ is abelian. Then, by the Kronecker-Weber theorem, there is an integer $n$ (which we choose to be minimal) such that $K\subseteq \mathbb{Q}({\zeta }_n)$ where ${\zeta }_n$ is a primitive $n$th root of unity. Let $G=\mathrm{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})\cong {(\mathbb{Z}/n\mathbb{Z})}^{\times }$ and let $\chi :G\to {ℂ}^{\times }$ be a Dirichlet character. Then the kernel of $\chi$ determines a fixed field of $\mathbb{Q}({\zeta }_n)$. Further, for any field $K$ as before, there exists a group $X$ of Dirichlet characters of $G$ such that $K$ is equal to the intersection of the fixed fields by the kernels of all $\chi \in X$. The order of $X$ is $[K:\mathbb{Q}]$ and $X\cong \mathrm{Gal}(K/\mathbb{Q})$.

Theorem ([1], Thm. 4.3).

Let $K$ be an abelian number field and let $X$ be the associated group of Dirichlet characters. The Dedekind zeta function of $K$ factors as follows:

$${\zeta }_K(s)=\prod _{\chi \in X}L(s,\chi ).$$

Notice that for the trivial character ${\chi }_0$ one has $L(s,{\chi }_0)=\zeta (s)$, the Riemann zeta function, which has a simple pole at $s=1$ with residue $1$. Thus, for an arbitrary abelian number field $K$:

$${\zeta }_K(s)=\prod _{\chi \in X}L(s,\chi )=\zeta (s)\cdot \prod _{{\chi }_0\ne \chi \in X}L(s,\chi )$$

where the last product is taken over all non-trivial characters $\chi \in X$.

Title	Factorization of the Dedekind zeta function of an abelian number field
Canonical name	FactorizationOfTheDedekindZetaFunctionOfAnAbelianNumberField
Date of creation	2013-03-22 16:01:21
Last modified on	2013-03-22 16:01:21
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Theorem
Classification	msc 11M06
Classification	msc 11R42
Related topic	ValuesOfDedekindZetaFunctionsOfRealQuadraticNumberFieldsAtNegativeIntegers