= Idelic local-global reciprocity equivalence
{title2=$\theta_{L/K}((a_v))=\prod_v\operatorname{rec}_v(a_v)$}
For a finite <abelian extension> of <number fields>, local reciprocity maps into the local <decomposition groups> combine to a continuous map on the <idele group>: outside finitely many places they kill units, and an idele has nonunit components only finitely often. The <Artin reciprocity law> is equivalent to the compatibility that this product kills every <principal idele>. Conversely, restrict a global continuous Artin homomorphism to each single-place embedding of $K_v^*$. These maps reconstruct the global map, by density of finite-component ideles and triviality on a basic open subgroup of units. Merely having unrelated local maps omits the essential principal-idele compatibility.
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