Cohomology of p-adic fields and Local class field theory

TL;DR

This paper provides an expository overview of the cohomological formulation of local class field theory, connecting Galois groups with Tate cohomology and norm groups in local fields.

Contribution

It presents a cohomological proof of the Local Reciprocity Law, linking Galois groups to Tate cohomology and norm groups in local fields.

Findings

Cohomological formulation of the Local Reciprocity Law

Connection between Galois groups and Tate cohomology

Application of homological algebra to local class field theory

Abstract

In this expository article, we outline a basic theory of group (co)homology and prove a cohomological formulation of the Local Reciprocity Law: $${\rm Gal}(L/K)^{\rm ab} \cong H_T^{-2}({\rm Gal}(L/K),\mathbb{Z}) \cong H_T^{0}({\rm Gal}(L/K),L^\times) \cong \frac{K^\times}{{\rm Nm}_{L/K}(L^\times)}$$ We first recall basic facts about local fields and homological algebra. Then we define group (co)homology, Tate cohomology, and furnish a toolbox. The Local Reciprocity Law is proven in an abstract cohomological setting, then applied to the case of local fields.