Idelic local-global reciprocity equivalence (${\theta }_{L/K}((a_v))={\prod }_v{\mathrm{rec}}_v(a_v)$)

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 $K_v^{\ast }$. 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.