Reciprocity law
mathematical law, a generalization of quadratic reciprocity
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials
f
(
x
)
{\displaystyle f(x)}
with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial
f
(
x
)
=
x
2
+
a
x
+
b
{\displaystyle f(x)=x^{2}+ax+b}
splits into linear terms when reduced mod
p
{\displaystyle p}
. That is, it determines for which prime numbers the relation
f
(
x
)
≡
f
p
(
x
)
=
(
x
−
n
p
)
(
x
−
m
p
)
(
mod
p
)
{\displaystyle f(x)\equiv f_{p}(x)=(x-n_{p})(x-m_{p}){\text{ }}({\text{mod }}p)}
holds. For a general reciprocity lawpg 3, it is defined as the rule determining which primes
p
{\displaystyle p}
the polynomial
f
p
{\displaystyle f_{p}}
splits into linear factors, denoted
Spl
{
f
(
x
)
}
{\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.
The name reciprocity law was coined by Legendre in his 1785 publication Recherches d'analyse indéterminée, because odd primes reciprocate or not in the sense of quadratic reciprocity stated below according to their residue classes
mod
4
{\displaystyle {\bmod {4}}}
.
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