Cohomology of p-adic fields and Local class field theory

In this expository article, we outline a basic theory of group (co)homology and prove a cohomological formulation of the Local Reciprocity Law: Gal(L/K)ᵃᵇ HT⁻²( Gal(L/K),Z) HT⁰( Gal(L/K),L^×) K^× NmL/K(L^×) 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.