  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.
