Power residue symbol

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 ${\mathcal {O}}_{k}$ that contains a primitive n-th root of unity $\zeta _{n}.$

Let $\mathfrak{p} \subset \mathcal{O}_k$ be a prime ideal and assume that n and ${\mathfrak {p}}$ are coprime (i.e. $n\not \in {\mathfrak {p}}$ .)

The norm of ${\mathfrak {p}}$ is defined as the cardinality of the residue class ring (note that since ${\mathfrak {p}}$ is prime the residue class ring is a finite field):

\mathrm {N} {\mathfrak {p}}:=|{\mathcal {O}}_{k}/{\mathfrak {p}}|.

An analogue of Fermat's theorem holds in $\mathcal{O}_k.$ If $\alpha \in {\mathcal {O}}_{k}-{\mathfrak {p}},$ then

\alpha ^{\mathrm {N} {\mathfrak {p}}-1}\equiv 1{\bmod {\mathfrak {p}}}.

And finally, $\mathrm {N} {\mathfrak {p}}\equiv 1{\bmod {n}}.$ These facts imply that

\alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}\equiv \zeta _{n}^{s}{\bmod {\mathfrak {p}}}

is well-defined and congruent to a unique $n$ -th root of unity $\zeta _{n}^{s}.$

Definition

This root of unity is called the n-th power residue symbol for ${\mathcal {O}}_{k},$ and is denoted by

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=\zeta _{n}^{s}\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}{\bmod {\mathfrak {p}}}.

Properties

The n-th power symbol has properties completely analogous to those of the classical (quadratic) Legendre symbol ( $\zeta$ is a fixed primitive $n$ -th root of unity):

\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)

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}{\bmod {\mathfrak {p}}}.

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}\left({\frac {\beta }{\mathfrak {p}}}\right)_{n}=\left({\frac {\alpha \beta }{\mathfrak {p}}}\right)_{n}

\alpha \equiv \beta {\bmod {\mathfrak {p}}}\quad \Rightarrow \quad \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=\left({\frac {\beta }{\mathfrak {p}}}\right)_{n}

Relation to the Hilbert symbol

The n-th power residue symbol is related to the Hilbert symbol $(\cdot ,\cdot )_{{{\mathfrak {p}}}}$ for the prime ${\mathfrak {p}}$ by

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=(\pi ,\alpha )_{\mathfrak {p}}

in the case ${\mathfrak {p}}$ coprime to n, where $\pi$ is any uniformising element for the local field $K_{{{\mathfrak {p}}}}$ .^[3]

Generalizations

The $n$ -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 ${\mathfrak {a}}\subset {\mathcal {O}}_{k}$ is the product of prime ideals, and in one way only:

{\mathfrak {a}}={\mathfrak {p}}_{1}\cdots {\mathfrak {p}}_{g}.

The $n$ -th power symbol is extended multiplicatively:

\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 $0\neq \beta \in {\mathcal {O}}_{k}$ then we define

\left({\frac {\alpha }{\beta }}\right)_{n}:=\left({\frac {\alpha }{(\beta )}}\right)_{n},

where $(\beta )$ is the principal ideal generated by $\beta .$

Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.

If $\alpha \equiv \beta {\bmod {\mathfrak {a}}}$ then $\left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}=\left({\tfrac {\beta }{\mathfrak {a}}}\right)_{n}.$
$\left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}\left({\tfrac {\beta }{\mathfrak {a}}}\right)_{n}=\left({\tfrac {\alpha \beta }{\mathfrak {a}}}\right)_{n}.$
$\left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}\left({\tfrac {\alpha }{\mathfrak {b}}}\right)_{n}=\left({\tfrac {\alpha }{\mathfrak {ab}}}\right)_{n}.$

Since the symbol is always an $n$ -th root of unity, because of its multiplicativity it is equal to 1 whenever one parameter is an $n$ -th power; the converse is not true.

If $\alpha \equiv \eta ^{n}{\bmod {\mathfrak {a}}}$ then $\left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}=1.$
If $\left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}\neq 1$ then $\alpha$ is not an $n$ -th power modulo ${\mathfrak {a}}.$
If $\left({\tfrac {\alpha }{\mathfrak {a}}}\right)_{n}=1$ then $\alpha$ may or may not be an $n$ -th power modulo ${\mathfrak {a}}.$

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]

\left({\frac {\alpha }{\beta }}\right)_{n}\left({\frac {\beta }{\alpha }}\right)_{n}^{-1}=\prod _{{\mathfrak {p}}|n\infty }(\alpha ,\beta )_{\mathfrak {p}},

whenever $\alpha$ and $\beta$ are coprime.

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, Berlin: Springer Science+Business Media, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002, doi:10.1007/978-3-662-12893-0
Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021

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