In algebraic number theory the n-th power residue symbol (for an integer n > 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
Let k be an algebraic number field with ring of integers
that contains a primitive n-th root of unity
Let
be a prime ideal and assume that n and
are coprime (i.e.
.)
The norm of
is defined as the cardinality of the residue class ring (note that since
is prime the residue class ring is a finite field):
![{\displaystyle \mathrm {N} {\mathfrak {p}}:=|{\mathcal {O}}_{k}/{\mathfrak {p}}|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7b7a4f22f422ff34cd6573e5a91e4c608f9779)
An analogue of Fermat's theorem holds in
If
then
![{\displaystyle \alpha ^{\mathrm {N} {\mathfrak {p}}-1}\equiv 1{\bmod {\mathfrak {p}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5595b7d12e199098ebaced73bcd7bd4915553d7b)
And finally, suppose
These facts imply that
![{\displaystyle \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}\equiv \zeta _{n}^{s}{\bmod {\mathfrak {p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2202a563f798e38e8ed387fbe8bcc9db4a2357)
is well-defined and congruent to a unique
-th root of unity
Definition
This root of unity is called the n-th power residue symbol for
and is denoted by
![{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=\zeta _{n}^{s}\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}{\bmod {\mathfrak {p}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe77061b0ea9e6f1b74e57853135547007b99483)
Properties
The n-th power symbol has properties completely analogous to those of the classical (quadratic) Jacobi symbol (
is a fixed primitive
-th root of unity):
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c509826ea6f805ec239204d545c630a92208213)
In all cases (zero and nonzero)
![{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}{\bmod {\mathfrak {p}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/163fa74f0f7fc7f938a620c63fbf9819237d8434)
![{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}\left({\frac {\beta }{\mathfrak {p}}}\right)_{n}=\left({\frac {\alpha \beta }{\mathfrak {p}}}\right)_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e20a1a52a89238e17e45e12754d89616d3ee6b15)
![{\displaystyle \alpha \equiv \beta {\bmod {\mathfrak {p}}}\quad \Rightarrow \quad \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=\left({\frac {\beta }{\mathfrak {p}}}\right)_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50a81798ffebb40325bc18fdc295115f2bac6550)
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
(the Carmichael lambda function of n).
Relation to the Hilbert symbol
The n-th power residue symbol is related to the Hilbert symbol
for the prime
by
![{\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=(\pi ,\alpha )_{\mathfrak {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/757a8c5efb3eef104341555d21a025dad007070e)
in the case
coprime to n, where
is any uniformising element for the local field
.[3]
Generalizations
The
-th power symbol may be extended to take non-prime ideals or non-zero elements as its "denominator", in the same way that the Jacobi symbol extends the Legendre symbol.
Any ideal
is the product of prime ideals, and in one way only:
![{\displaystyle {\mathfrak {a}}={\mathfrak {p}}_{1}\cdots {\mathfrak {p}}_{g}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f75153e37e5e38740ae00985010152c27106d6aa)
The
-th power symbol is extended multiplicatively:
![{\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/717d1c428c00fda54c4a4d397efaa55ed0ed0200)
For
then we define
![{\displaystyle \left({\frac {\alpha }{\beta }}\right)_{n}:=\left({\frac {\alpha }{(\beta )}}\right)_{n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4bbe32502eb407399af20feeef5ffb5afb62f79)
where
is the principal ideal generated by
Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.
- If
then ![{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}=\left({\tfrac {\beta }{\mathfrak {a}}}\right)_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/184e3b2dbeaa4da1cb8a73497fc6e7277de2d55e)
![{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}\left({\tfrac {\beta }{\mathfrak {a}}}\right)_{n}=\left({\tfrac {\alpha \beta }{\mathfrak {a}}}\right)_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08ccfe5ecbd753ee263db8f269377d4381cccefe)
![{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}\left({\tfrac {\alpha }{\mathfrak {b}}}\right)_{n}=\left({\tfrac {\alpha }{\mathfrak {ab}}}\right)_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f41fe2842c654d79cd0413b7d1a379fc0deb21a8)
Since the symbol is always an
-th root of unity, because of its multiplicativity it is equal to 1 whenever one parameter is an
-th power; the converse is not true.
- If
then ![{\displaystyle \left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a95cd03dd22c7cdd7e142883055391c302f800d)
- If
then
is not an
-th power modulo ![{\displaystyle {\mathfrak {a}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dec6f4cf3b8392c15fc18c4e149f882d4e0182d)
- If
then
may or may not be an
-th power modulo ![{\displaystyle {\mathfrak {a}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dec6f4cf3b8392c15fc18c4e149f882d4e0182d)
Power reciprocity law
The power reciprocity law, the analogue of the law of quadratic reciprocity, may be formulated in terms of the Hilbert symbols as[4]
![{\displaystyle \left({\frac {\alpha }{\beta }}\right)_{n}\left({\frac {\beta }{\alpha }}\right)_{n}^{-1}=\prod _{{\mathfrak {p}}|n\infty }(\alpha ,\beta )_{\mathfrak {p}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0c9475fdf8a765e0c620d50a7dd1427c54b6cb)
whenever
and
are coprime.
See also
Notes
- ^ 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
References
- 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