On the Analytic Continuation of the Dedekind Zeta Function of Abelian Galois Number Fields

On the Analytic Continuation of the Dedekind Zeta Function of Abelian Galois Number Fields

Pure Mathematics

Samay Varjangbhay

Dr Alvaro Gonzalez Hernandez

The aim of this article is to prove the analytic continuation and functional equation of the Dedekind zeta function associated with abelian Galois number fields. The method used intertwines Fourier analysis, Artin decomposition, and Mellin transforms of Dirichlet L-functions. We first establish the functional equation for a general Dirichlet L-function, using theta functions and Poisson summation. Moreover, applying Artin decomposition to cyclotomic fields gives the analytic continuation of the Dedekind zeta function for prime cyclotomic extensions. We then use the Kronecker-Weber theorem about abelian field extensions to extend this method to work for abelian number fields.

Algebraic Number Theory, Representation Theory, Galois Theory