Local Class Field Theory

Abstract

Class Field Theory is a very powerful tool for answering some of the fundamental questions in Algebraic Number Theory like splitting of primes and reciprocity laws. This thesis is an attempt to understand Local Class Field Theory through

the cohomological approach. It includes a deep study of the standard group co- homology as well as the Tate Cohomology. Then we describe how this machinery

can be utilized to classify the finite abelian extensions of a local field in terms of norm subgroups. We also explain why this approach cannot be extended to

classify the non-abelian extensions. The thesis is intended to be a liberal intro- duction to Local Class Field Theory as very detailed reasonings are provided

and not much background is assumed apart from the knowledge of standard material covered in graduate courses.