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 $\operatorname{Gal}(K/\mathbb{Q})$ 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 $G=\operatorname{Gal}(\mathbb{Q}(\zeta_{n})/\mathbb{Q})\cong(\mathbb{Z}/n% \mathbb{Z})^{\times}$ and let $\chi:G\to\mathbb{C}^{\times}$ 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 $X\cong\operatorname{Gal}(K/\mathbb{Q})$.

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 $K$:

 $\zeta_{K}(s)=\prod_{\chi\in X}L(s,\chi)=\zeta(s)\cdot\prod_{\chi_{0}\neq\chi% \in X}L(s,\chi)$

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

References

Title Factorization of the Dedekind zeta function of an abelian number field FactorizationOfTheDedekindZetaFunctionOfAnAbelianNumberField 2013-03-22 16:01:21 2013-03-22 16:01:21 alozano (2414) alozano (2414) 4 alozano (2414) Theorem msc 11M06 msc 11R42 ValuesOfDedekindZetaFunctionsOfRealQuadraticNumberFieldsAtNegativeIntegers