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Mathematical law, a generalization of quadratic reciprocity
various other concepts of reciprocity and reciprocity laws

In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial splits into linear terms when reduced mod That is, it determines for which prime numbers the relationholds. For a general reciprocity lawpg 3, it is defined as the rule determining which primes the polynomial splits into linear factors, denoted
There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols (a,b/p), taking values in roots of unity, is equal to 1. Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K groups, and their relationship with the original quadratic reciprocity law can be hard to see. The name reciprocity law was coined by Legendre in his 1785 publication Recherches d'analyse indéterminée, because odd primes reciprocate or not in the sense of quadratic reciprocity stated below according to their residue classes This reciprocating behavior does not generalize well, the equivalent splitting behavior does. The name reciprocity law is still used in the more general context of splittings.

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 .

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.

A rational reciprocity law is one stated in terms of rational integers without the use of roots of unity.

The Langlands program includes several conjectures for general reductive algebraic groups, which for the special of the group GL1 imply the Artin reciprocity law.

Yamamoto's reciprocity law is a reciprocity law related to class numbers of quadratic number fields.