# Reciprocity law

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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[1]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.

In terms of the Legendre symbol, the law of quadratic reciprocity for positive odd primes states

${\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}$

## 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

${\displaystyle {\Bigg (}{\frac {\alpha }{\beta }}{\Bigg )}_{3}={\Bigg (}{\frac {\beta }{\alpha }}{\Bigg )}_{3}.}$

## 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

${\displaystyle {\Bigg [}{\frac {\pi }{\theta }}{\Bigg ]}\left[{\frac {\theta }{\pi }}\right]^{-1}=(-1)^{{\frac {N\pi -1}{4}}{\frac {N\theta -1}{4}}}.}$

## Eisenstein reciprocity

Suppose that ζ is an ${\displaystyle l}$th root of unity for some odd prime ${\displaystyle l}$. The power character is the power of ζ such that

${\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{l}\equiv \alpha ^{\frac {N({\mathfrak {p}})-1}{l}}{\pmod {\mathfrak {p}}}}$

for any prime ideal ${\displaystyle {\mathfrak {p}}}$ of Z[ζ]. It is extended to other ideals by multiplicativity. The Eisenstein reciprocity law states that

${\displaystyle \left({\frac {a}{\alpha }}\right)_{l}=\left({\frac {\alpha }{a}}\right)_{l}}$

for a any rational integer coprime to ${\displaystyle l}$ and α any element of Z[ζ] that is coprime to a and ${\displaystyle l}$ and congruent to a rational integer modulo (1–ζ)2.

## Kummer reciprocity

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

${\displaystyle \left\{{\frac {p}{q}}\right\}^{n}=\left\{{\frac {p^{n}}{q}}\right\}}$ where n is some integer prime to l such that pn is principal.

The Kummer reciprocity law states that

${\displaystyle \left\{{\frac {p}{q}}\right\}=\left\{{\frac {q}{p}}\right\}}$

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

${\displaystyle \prod _{v}(a,b)_{v}=1}$

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 ${\displaystyle (p,q)_{\infty }(p,q)_{2}(p,q)_{p}(p,q)_{q}=1}$ 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

${\displaystyle \theta :C_{K}/{N_{L/K}(C_{L})}\to {\text{Gal}}(L/K)^{\text{ab}}.}$

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[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 ${\displaystyle K^{\times }/N_{L/K}(L^{\times })}$ onto the Galois group ${\displaystyle Gal(L/K)}$.

## 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[2]

${\displaystyle \left({\frac {\alpha }{\beta }}\right)_{n}\left({\frac {\beta }{\alpha }}\right)_{n}^{-1}=\prod _{{\mathfrak {p}}|n\infty }(\alpha ,\beta )_{\mathfrak {p}}\ .}$

## Rational reciprocity laws

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

## Langlands reciprocity

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

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