Artin Reciprocity
In the language if ideles, the Artin reciprocity law for a finite extension L/K state that the Artin map from the idele class group CK to the abelianization Gal(L/K)ab of the Galois group vanishes on NL/K(CL), and induces an isomorphism
Although it is not immediately obvious, the Artin reciprocity law easily implies all the previously discovered reciprocity laws, by applying it to suitable extensions L/K. For example, in the special case when K contains the nth roots of unity and L=K is a Kummer extension of K, the fact that the Artin map vanishes on NL/K(CL) implies Hilbert's reciprocity law for the Hilbert symbol.
Read more about this topic: Reciprocity Law
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