Reciprocity law (mathematics)
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity.
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.
In terms of the Legendre symbol, the law of quadratic reciprocity for positive odd primes states
The law of cubic reciprocity for Eisenstein integers states that if α and β are primary (primes congruent to 2 mod 3) then
In terms of the quartic residue symbol, the law of quartic reciprocity for Gaussian integers states that if π and θ are primary (congruent to 1 mod (1+i)3) Gaussian primes then
Suppose that ζ is an th root of unity for some odd prime . The power character is the power of ζ such that
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