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.
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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.
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