Factorization of the Dedekind zeta function of an abelian number fieldFactorizationOfTheDedekindZetaFunctionOfAnAbelianNumberField2006-06-20 13:29:342006-06-20 13:29:34TheoremDedekind zeta function% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}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. $K/\Rats$ is Galois and
$\Gal(K/\Rats)$ is abelian. Then, by the Kronecker-Weber theorem,
there is an integer $n$ (which we choose to be minimal) such that
$K\subseteq \Rats(\zeta_n)$ where $\zeta_n$ is a primitive $n$th
root of unity. Let $G=\Gal(\Rats(\zeta_n)/\Rats)\cong
(\Ints/n\Ints)^\times$ and let $\chi:G\to \Complex^\times$ be a
Dirichlet character. Then the kernel of $\chi$ determines a fixed
field of $\Rats(\zeta_n)$. 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 $[K:\Rats]$ and
$X\cong \Gal(K/\Rats)$.
\begin{thm}[\cite{wash}, Thm. 4.3]
\label{factor} 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:
$$\zeta_K(s)=\prod_{\chi \in X} L(s,\chi).$$
\end{thm}
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 $\chi \in X$.
\begin{thebibliography}{99}
\bibitem{wash} L. C. Washington, {\em Introduction to Cyclotomic Fields}, Springer-Verlag, New York.
\end{thebibliography}