Class Field Theory

Abstract

Class Field Theory gives a one-one correspondence between the Galois groups of finite abelian extensions of a global field, k, and open subgroups of finite index in class group. This correspondence is captured by Reciprocity map and Existence theorem. We first derive these theorems for local fields using Tate’s theorem and Lubin-Tate Formal groups. From local case we go to global case using cohomology of Adeles and Ideles.