In algebraic number theory the n -th power residue symboln > 2) is a generalization of the (quadratic) Legendre symbol to n -th powers. These symbols are used in the statement and proof of cubic , quartic , Eisenstein , and related higher[ 1] reciprocity laws .[ 2]
Background and notation [ edit ] Let k be an algebraic number field with ring of integers
O
k
{\displaystyle {\mathcal {O}}_{k}}
primitive n -th root of unity
ζ
n
.
{\displaystyle \zeta _{n}.}
Let
p
⊂
O
k
{\displaystyle {\mathfrak {p}}\subset {\mathcal {O}}_{k}}
prime ideal and assume that n and
p
{\displaystyle {\mathfrak {p}}}
coprime (i.e.
n
∉
p
{\displaystyle n\not \in {\mathfrak {p}}}
The norm of
p
{\displaystyle {\mathfrak {p}}}
p
{\displaystyle {\mathfrak {p}}}
finite field ):
N
p
:=
|
O
k
/
p
|
.
{\displaystyle \mathrm {N} {\mathfrak {p}}:=|{\mathcal {O}}_{k}/{\mathfrak {p}}|.}
An analogue of Fermat's theorem holds in
O
k
.
{\displaystyle {\mathcal {O}}_{k}.}
α
∈
O
k
−
p
,
{\displaystyle \alpha \in {\mathcal {O}}_{k}-{\mathfrak {p}},}
α
N
p
−
1
≡
1
mod
p
.
{\displaystyle \alpha ^{\mathrm {N} {\mathfrak {p}}-1}\equiv 1{\bmod {\mathfrak {p}}}.}
And finally, suppose
N
p
≡
1
mod
n
.
{\displaystyle \mathrm {N} {\mathfrak {p}}\equiv 1{\bmod {n}}.}
α
N
p
−
1
n
≡
ζ
n
s
mod
p
{\displaystyle \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}\equiv \zeta _{n}^{s}{\bmod {\mathfrak {p}}}}
is well-defined and congruent to a unique
n
{\displaystyle n}
ζ
n
s
.
{\displaystyle \zeta _{n}^{s}.}
This root of unity is called the n -th power residue symbol for
O
k
,
{\displaystyle {\mathcal {O}}_{k},}
(
α
p
)
n
=
ζ
n
s
≡
α
N
p
−
1
n
mod
p
.
{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=\zeta _{n}^{s}\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}{\bmod {\mathfrak {p}}}.}
The n -th power symbol has properties completely analogous to those of the classical (quadratic) Jacobi symbol (
ζ
{\displaystyle \zeta }
n
{\displaystyle n}
(
α
p
)
n
=
{
0
α
∈
p
1
α
∉
p
and
∃
η
∈
O
k
:
α
≡
η
n
mod
p
ζ
α
∉
p
and there is no such
η
{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}={\begin{cases}0&\alpha \in {\mathfrak {p}}\\1&\alpha \not \in {\mathfrak {p}}{\text{ and }}\exists \eta \in {\mathcal {O}}_{k}:\alpha \equiv \eta ^{n}{\bmod {\mathfrak {p}}}\\\zeta &\alpha \not \in {\mathfrak {p}}{\text{ and there is no such }}\eta \end{cases}}}
In all cases (zero and nonzero)
(
α
p
)
n
≡
α
N
p
−
1
n
mod
p
.
{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}{\bmod {\mathfrak {p}}}.}
(
α
p
)
n
(
β
p
)
n
=
(
α
β
p
)
n
{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}\left({\frac {\beta }{\mathfrak {p}}}\right)_{n}=\left({\frac {\alpha \beta }{\mathfrak {p}}}\right)_{n}}
α
≡
β
mod
p
⇒
(
α
p
)
n
=
(
β
p
)
n
{\displaystyle \alpha \equiv \beta {\bmod {\mathfrak {p}}}\quad \Rightarrow \quad \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=\left({\frac {\beta }{\mathfrak {p}}}\right)_{n}}
All power residue symbols mod n are Dirichlet characters mod n , and the m -th power residue symbol only contains the m -th roots of unity , the m -th power residue symbol mod n exists if and only if m divides
λ
(
n
)
{\displaystyle \lambda (n)}
Carmichael lambda function of n ).
Relation to the Hilbert symbol [ edit ] The n -th power residue symbol is related to the Hilbert symbol
(
⋅
,
⋅
)
p
{\displaystyle (\cdot ,\cdot )_{\mathfrak {p}}}
p
{\displaystyle {\mathfrak {p}}}
(
α
p
)
n
=
(
π
,
α
)
p
{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=(\pi ,\alpha )_{\mathfrak {p}}}
in the case
p
{\displaystyle {\mathfrak {p}}}
n , where
π
{\displaystyle \pi }
uniformising element for the local field
K
p
{\displaystyle K_{\mathfrak {p}}}
[ 3]
The
n
{\displaystyle n}
Jacobi symbol extends the Legendre symbol.
Any ideal
a
⊂
O
k
{\displaystyle {\mathfrak {a}}\subset {\mathcal {O}}_{k}}
a
=
p
1
⋯
p
g
.
{\displaystyle {\mathfrak {a}}={\mathfrak {p}}_{1}\cdots {\mathfrak {p}}_{g}.}
The
n
{\displaystyle n}
(
α
a
)
n
=
(
α
p
1
)
n
⋯
(
α
p
g
)
n
.
{\displaystyle \left({\frac {\alpha }{\mathfrak {a}}}\right)_{n}=\left({\frac {\alpha }{{\mathfrak {p}}_{1}}}\right)_{n}\cdots \left({\frac {\alpha }{{\mathfrak {p}}_{g}}}\right)_{n}.}
For
0
≠
β
∈
O
k
{\displaystyle 0\neq \beta \in {\mathcal {O}}_{k}}
(
α
β
)
n
:=
(
α
(
β
)
)
n
,
{\displaystyle \left({\frac {\alpha }{\beta }}\right)_{n}:=\left({\frac {\alpha }{(\beta )}}\right)_{n},}
where
(
β
)
{\displaystyle (\beta )}
β
.
{\displaystyle \beta .}
Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.
If
α
≡
β
mod
a
{\displaystyle \alpha \equiv \beta {\bmod {\mathfrak {a}}}}
(
α
a
)
n
=
(
β
a
)
n
.
{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}=\left({\tfrac {\beta }{\mathfrak {a}}}\right)_{n}.}
(
α
a
)
n
(
β
a
)
n
=
(
α
β
a
)
n
.
{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}\left({\tfrac {\beta }{\mathfrak {a}}}\right)_{n}=\left({\tfrac {\alpha \beta }{\mathfrak {a}}}\right)_{n}.}
(
α
a
)
n
(
α
b
)
n
=
(
α
a
b
)
n
.
{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}\left({\tfrac {\alpha }{\mathfrak {b}}}\right)_{n}=\left({\tfrac {\alpha }{\mathfrak {ab}}}\right)_{n}.}
Since the symbol is always an
n
{\displaystyle n}
n
{\displaystyle n}
If
α
≡
η
n
mod
a
{\displaystyle \alpha \equiv \eta ^{n}{\bmod {\mathfrak {a}}}}
(
α
a
)
n
=
1.
{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}=1.}
If
(
α
a
)
n
≠
1
{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}\neq 1}
α
{\displaystyle \alpha }
n
{\displaystyle n}
a
.
{\displaystyle {\mathfrak {a}}.}
If
(
α
a
)
n
=
1
{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}=1}
α
{\displaystyle \alpha }
n
{\displaystyle n}
a
.
{\displaystyle {\mathfrak {a}}.}
Power reciprocity law [ edit ] The power reciprocity law , the analogue of the law of quadratic reciprocity , may be formulated in terms of the Hilbert symbols as[ 4]
(
α
β
)
n
(
β
α
)
n
−
1
=
∏
p
|
n
∞
(
α
,
β
)
p
,
{\displaystyle \left({\frac {\alpha }{\beta }}\right)_{n}\left({\frac {\beta }{\alpha }}\right)_{n}^{-1}=\prod _{{\mathfrak {p}}|n\infty }(\alpha ,\beta )_{\mathfrak {p}},}
whenever
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
^ Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers.^ All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2
^ Neukirch (1999) p. 336
^ Neukirch (1999) p. 415
Gras, Georges (2003), Class field theory. From theory to practice , Springer Monographs in Mathematics, Berlin: Springer-Verlag , pp. 204–207, ISBN 3-540-44133-6 Zbl 1019.11032 Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition) , New York: Springer Science+Business Media , ISBN 0-387-97329-X Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein , Springer Monographs in Mathematics, Berlin: Springer Science+Business Media , doi :10.1007/978-3-662-12893-0 , ISBN 3-540-66957-4 MR 1761696 , Zbl 0949.11002 Neukirch, Jürgen (1999), Algebraic number theory , Grundlehren der Mathematischen Wissenschaften, vol. 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag , ISBN 3-540-65399-6 Zbl 0956.11021 
