Cohomology of p-adic fields and 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: $$\mathrm{Gal}(L/K{)}^{\mathrm{ab}}\cong H_T^{-2}(\mathrm{Gal}(L/K),\mathbb{Z})\cong H_T^0(\mathrm{Gal}(L/K),L^{\times })\cong \frac{K^{\times }}{{\mathrm{Nm}}_{L/K}(L^{\times })}$$ 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.