Class field theory - Artin reciprocity law

Abstract

In this Thesis we give an introduction in Class Field Theory, provingArtin reciprocity law. The goal of class field theory is to describe the Galois extensions of a local or global field in terms of the arithmetic of the field itself. Apart from a few remarks about the more general cases, these notes will concentrate on the case of abelian extensions, which is the basic case. We give the framework of the theoryintroducing Abstract class field theory and we can see how this canbe translated in the case of global class field theory using idele classgroups as modules or multiplicative groups in the case of local classfield theory. The language that we use is purely algebraic, with theexception of an analytic approach which is mostly redundant nowadays after much effort of the pioneers in that field to confront such a defect, as it was considered.