in a unique way such that where n is some integer prime to l such that pn is principal. The Kummer reciprocity law states that for p and q any distinct prime ideals of Z[ζ] other than (1–ζ). Hilbert reciprocity In terms of the Hilbert symbol, Hilbert's reciprocity law for an algebraic number field states that where the product is over all finite and infinite places. Over the rational numbers this is equivalent to the law of quadratic reciprocity. To see this take a and b to be distinct odd primes. Then Hilbert's law becomes But (p,q)p is equal to the Legendre symbol, (p,q)∞ is 1 if one of p and q is positive and –1 otherwise, and (p,q)2 is (–1)(p–1)(q–1)/4. So for p and q positive odd primes Hilbert's law is the law of quadratic reciprocity. Artin reciprocity In the language of ideles, the Artin reciprocity law for a finite extension L/K states 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[a1/n] 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. Local reciprocity Hasse introduced a local analogue of the Artin reciprocity law, called the local reciprocity law. One form of it states that for a finite abelian extension of L/K of local fields, the Artin map is an isomorphism from onto the Galois group . Explicit reciprocity laws In order to get a classical style reciprocity law from the Hilbert reciprocity law Π(a,b)p=1, one needs to know the values of (a,b)p for p dividing n. Explicit formulas for this are sometimes called explicit reciprocity laws. Power reciprocity laws A power reciprocity law may be formulated as an analogue of the law of quadratic reciprocity in terms of the Hilbert symbols as Rational reciprocity laws A rational reciprocity law is one stated in terms of rational integers without the use of roots of unity. Scholz's reciprocity law Shimura reciprocity Weil reciprocity law Langlands reciprocity The Langlands program includes several conjectures for general reductive algebraic groups, which for the