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
be an abelian number field, i.e. is Galois and
is abelian. Then, by the Kronecker-Weber theorem,
there is an integer (which we choose to be minimal) such that
where is a primitive th
root of unity
. Let and let be a
Dirichlet character
. Then the kernel of determines a fixed
field of . 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 . The order of is and
.
Theorem ([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:
Notice that for the trivial character one has
, the Riemann zeta function, which has a
simple pole
at with residue
. Thus, for an arbitrary
abelian number field :
where the last product is taken over all non-trivial characters .
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 |