Reciprocity law

In terms of the Legendre symbol, the law of quadratic reciprocity states

for positive odd primes ${\displaystyle p,q}$ we have ${\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}$

Using the definition of the Legendre symbol this is equivalent to a more elementary statement about equations.

For positive odd primes ${\displaystyle p,q}$ the solubility of ${\displaystyle n^{2}-p\equiv 0{\bmod {q}}}$ for ${\displaystyle n}$ determines the solubility of ${\displaystyle m^{2}-q\equiv 0{\bmod {p}}}$ for ${\displaystyle m}$ and vice versa by the comparatively simple criterion whether ${\displaystyle (-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}}$ is ${\displaystyle 1}$ or ${\displaystyle -1}$.

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 ${\displaystyle p,q}$ the splitting of the polynomial ${\displaystyle x^{2}-p}$ in ${\displaystyle {\bmod {q}}}$-residues determines the splitting of the polynomial ${\displaystyle x^{2}-q}$ in ${\displaystyle {\bmod {p}}}$-residues and vice versa through the quantity ${\displaystyle (-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}\in \{\pm 1\}}$.

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.

The law of cubic reciprocity for Eisenstein integers states that if α and β are primary (primes congruent to 2 mod 3) then

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

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

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

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.

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).

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.

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 C_K to the abelianization Gal(L/K)^ab of the Galois group vanishes on N_L/K(C_L), 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[a^1/n] is a Kummer extension of K, the fact that the Artin map vanishes on N_L/K(C_L) 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 ${\displaystyle K^{\times }/N_{L/K}(L^{\times })}$ onto the Galois group ${\displaystyle Gal(L/K)}$.

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]

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

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.

• Hilbert's ninth problem
• Stanley's reciprocity theorem

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• Lemmermeyer, Franz, Reciprocity laws. From Kummer to Hilbert
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• Stepanov, S. A. (2001) [1994], "Reciprocity laws", Encyclopedia of Mathematics, EMS Press
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