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: $$\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}K{)}^{\mathrm{a}\mathrm{b}}\cong H_T^{-2}(\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}K),\mathbb{Z})\cong H_T^0(\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}K),L^{\times })\cong \frac{K^{\times }}{{\mathrm{N}\mathrm{m}}_{L\mathrm{/}K}(L^{\times })}$$ 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.