= Cup-product construction of local reciprocity
{title2=$G^{\mathrm{ab}}\xrightarrow{\ \smile u_{L/K}\ }K^\times/N_{L/K}(L^\times)$}
For $G=\operatorname{Gal}(L/K)$, <Tate cohomology of a finite group> identifies the source as $\widehat H^{-2}(G,\mathbb Z)$ and the target as $\widehat H^0(G,L^\times)$. Cup product with the <local fundamental class> is an isomorphism; its inverse is the finite <Local Artin map>. The algebraic cup-product isomorphism applies because every subgroup has first cohomology zero by <Hilbert theorem 90>, and second cohomology cyclic of order equal to its size by the <local Brauer invariant>. Compatibility with restriction and corestriction yields the tower and norm compatibility of <Local Artin reciprocity>.
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