Class Field Theory

For a local or global field $KK$, abelian extensions of $KK$ are classified by the arithmetic of $K^{\times }K^\backslash times$ or $A_K^{\times }/K^{\times }\backslash mathbb\{A\}\_K^\backslash times/K^\backslash times$, yielding reciprocity isomorphisms such as $\mathrm{Gal}\left(K^{ab}/K\right)\cong K^{\times }{}^{\wedge }\backslash operatorname\{Gal\}(K^\{\backslash mathrm\{ab\}\}/K)\; \backslash cong\; K^\backslash times\{\}^\{\backslash wedge\}$ in the local case and its idele-class analogue globally.