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 (, 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:
where the last product is taken over all non-trivial characters .
- 1 L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.
|Title||Factorization of the Dedekind zeta function of an abelian number field|
|Date of creation||2013-03-22 16:01:21|
|Last modified on||2013-03-22 16:01:21|
|Last modified by||alozano (2414)|