In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials ${\displaystyle f(x)}$ with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial ${\displaystyle f(x)=x^{2}+ax+b}$ splits into linear terms when reduced mod ${\displaystyle p}$. That is, it determines for which prime numbers the relation

${\displaystyle f(x)\equiv f_{p}(x)=(x-n_{p})(x-m_{p}){\text{ }}({\text{mod }}p)}$

holds. For a general reciprocity law^{pg 3}, it is defined as the rule determining which primes ${\displaystyle p}$ the polynomial ${\displaystyle f_{p}}$ splits into linear factors, denoted ${\displaystyle {\text{Spl}}\{f(x)\}}$.

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.