In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials
f(x)
f(x)=x2+ax+b
p
holds. For a general reciprocity law[1] pg 3, it is defined as the rule determining which primesf(x)\equivfp(x)=(x-np)(x-mp)(modp)
p
fp
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.
The name reciprocity law was coined by Legendre in his 1785 publication Recherches d'analyse indéterminée,[2] because odd primes reciprocate or not in the sense of quadratic reciprocity stated below according to their residue classes
\bmod4
See main article: quadratic reciprocity. In terms of the Legendre symbol, the law of quadratic reciprocity states for positive odd primes
p,q
| \left( | p |
| q |
\right)\left(
| q | |
| p |
\right)=
| ||||||||
| (-1) |
.
Using the definition of the Legendre symbol this is equivalent to a more elementary statement about equations. For positive odd primes
p,q
n2-p\equiv0\bmodq
n
m2-q\equiv0\bmodp
m
| ||||||||
| (-1) |
1
-1
By the factor theorem and the behavior of degrees in factorizations the solubility of such quadratic congruence equations is equivalent to the splitting of associated quadratic polynomials over a residue ring into linear factors. In this terminology the law of quadratic reciprocity is stated as follows. For positive odd primes
p,q
x2-p
\bmodq
x2-q
\bmodp
| ||||||||
| (-1) |
\in\{\pm1\}
This establishes the bridge from the name giving reciprocating behavior of primes introduced by Legendre to the splitting behavior of polynomials used in the generalizations.
See main article: cubic reciprocity. The law of cubic reciprocity for Eisenstein integers states that if α and β are primary (primes congruent to 2 mod 3) then
| ( | \alpha |
| \beta |
)3=(
| \beta | |
| \alpha |
)3.
See main article: quartic reciprocity. 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
| [ | \pi | ]\left[ |
| \theta |
| \theta | |
| \pi |
\right]-1
| ||||||||
| = (-1) |
.
See main article: Octic reciprocity.
See main article: Eisenstein reciprocity. Suppose that ζ is an
l
l
| \left( | \alpha |
| ak{p |
ak{p}
| \left( | a |
| \alpha |
| \right) | ||||
|
\right)l
l
l
Suppose that ζ is an lth root of unity for some odd regular prime l. Since l is regular, we can extend the symbol to ideals in a unique way such that
| \left\{ | p |
| q |
| ||||
| \right\} |
\right\}
| \left\{ | p | \right\}=\left\{ |
| q |
| q | |
| p |
\right\}
See main article: Hilbert symbol. In terms of the Hilbert symbol, Hilbert's reciprocity law for an algebraic number field states that
\prodv(a,b)v=1
(p,q)infty(p,q)2(p,q)p(p,q)q=1
See main article: Artin reciprocity law. 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
\theta:CK/{NL/K(CL)}\toGal(L/K)ab.
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[''a''<sup>1/''n''</sup>] 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.
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
K x /NL/K(L x )
Gal(L/K)
See main article: Explicit reciprocity law.
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 power reciprocity law may be formulated as an analogue of the law of quadratic reciprocity in terms of the Hilbert symbols as[3]
| \left({ | \alpha |
| \beta |
See main article: Rational reciprocity law. A rational reciprocity law is one stated in terms of rational integers without the use of roots of unity.
See main article: Scholz's reciprocity law.
See main article: Shimura's reciprocity law.
See main article: Weil reciprocity law.
The Langlands program includes several conjectures for general reductive algebraic groups, which for the special of the group GL1 imply the Artin reciprocity law.
See main article: Yamamoto's reciprocity law. Yamamoto's reciprocity law is a reciprocity law related to class numbers of quadratic number fields.