Reciprocity law

In terms of the Legendre symbol, the law of quadratic reciprocity states for positive odd primes p,q we have \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 p,q the solubility of n^2-p\equiv0\bmod q for n determines the solubility of m^2-q\equiv0\bmod p for m and vice versa by the comparatively simple criterion whether (-1)^{\frac{p-1}{2}\frac{q-1}{2}} is 1 or -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 the splitting of the polynomial x^2-p in \bmod q-residues determines the splitting of the polynomial x^2-q in \bmod p-residues and vice versa through the quantity (-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. ==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 :\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 :\Bigg[\frac{\pi}{\theta}\Bigg]\left[\frac{\theta}{\pi}\right]^{-1}= (-1)^{\frac{N\pi - 1}{4}\frac{N\theta-1}{4}}. ==Octic reciprocity==

Suppose that ζ is an lth root of unity for some odd prime l. The power character is the power of ζ such that :\left(\frac{\alpha}{\mathfrak{p}}\right)_l \equiv \alpha^{\frac{N(\mathfrak{p})-1}{l}} \pmod {\mathfrak{p}} for any prime ideal \mathfrak{p} of Z[ζ]. It is extended to other ideals by multiplicativity. The Eisenstein reciprocity law states that : \left(\frac{a}{\alpha}\right)_l=\left(\frac{\alpha}{a}\right)_l for a any rational integer coprime to l and α any element of Z[ζ] that is coprime to a and 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 : \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 : \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 :\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 (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 : \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 K^{\times}/N_{L/K}(L^{\times}) onto the Galois group 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 :\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. ==Scholz's 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. ==Yamamoto's reciprocity law==

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