PlanetMath: Factorization of the Dedekind zeta function of an abelian number field

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Factorization of the Dedekind zeta function of an abelian number field

(Theorem)

The Dedekind zeta function of an abelian number field factors as a product of Dirichlet L-functions as follows. Let 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 (which we choose to be minimal) such that $K\subseteq \mathbb{Q}(\zeta_n)$ where $\zeta_n$ is a primitive 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 as before, there exists a group of Dirichlet characters of such that is equal to the intersection of the fixed fields by the kernels of all $\chi\in X$ . The order of is $[K:\mathbb{Q}]$ and $X\cong \operatorname{Gal}(K/\mathbb{Q})$ .

Theorem 1 ([1], Thm. 4.3) Let be an abelian number field and let be the associated group of Dirichlet characters. The Dedekind zeta function of factors as follows:

$\displaystyle \zeta_K(s)=\prod_{\chi \in X} L(s,\chi).$

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

with residue

. Thus, for an arbitrary abelian number field

$\displaystyle \zeta_K(s)=\prod_{\chi \in X}L(s,\chi)=\zeta(s)\cdot \prod_{\chi_0\neq \chi \in X} L(s,\chi)$

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

Bibliography

1: L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.

"Factorization of the Dedekind zeta function of an abelian number field" is owned by alozano.

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Cross-references: characters, residue, simple pole, Riemann zeta function, trivial character, order, intersection, group, field, fixed field, kernel, Dirichlet character, root of unity, primitive, minimal, integer, Kronecker-Weber theorem, abelian, product, factors, abelian number field, Dedekind zeta function
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This is version 1 of Factorization of the Dedekind zeta function of an abelian number field, born on 2006-06-20.
Object id is 8062, canonical name is FactorizationOfTheDedekindZetaFunctionOfAnAbelianNumberField.
Accessed 822 times total.

Classification:

AMS MSC:	11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $)
	11R42 (Number theory :: Algebraic number theory: global fields :: Zeta functions and $L$-functions of number fields)

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