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 $\chi :G\to {\u2102}^{\times}$ 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 $\chi \in X$.
References
- 1 L. C. Washington, Introduction to Cyclotomic Fields^{}, Springer-Verlag, New York.
Title | Factorization of the Dedekind zeta function of an abelian number field |
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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 |