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

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

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

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

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

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

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

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

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

Definition

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

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

Properties

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

${\displaystyle \left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}={\begin{cases}0&\alpha \in {\mathfrak {p}}\\1&\alpha \not \in {\mathfrak {p}}{\text{ and }}\exists \eta \in {\mathcal {O}}_{k}:\alpha \equiv \eta ^{n}{\bmod {\mathfrak {p}}}\\\zeta &\alpha \not \in {\mathfrak {p}}{\text{ and there is no such }}\eta \end{cases}}}$

In all cases (zero and nonzero)

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

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

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

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 ${\displaystyle \lambda (n)}$ (the Carmichael lambda function of n).

Relation to the Hilbert symbol

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

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

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

Generalizations

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

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

The ${\displaystyle n}$-th power symbol is extended multiplicatively:

${\displaystyle \left({\frac {\alpha }{\mathfrak {a}}}\right)_{n}=\left({\frac {\alpha }{{\mathfrak {p}}_{1}}}\right)_{n}\cdots \left({\frac {\alpha }{{\mathfrak {p}}_{g}}}\right)_{n}.}$

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

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

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

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

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

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

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

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]

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

whenever ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are coprime.

See also

• Modular_arithmetic#Residue_class
• Quadratic_residue#Prime_power_modulus
• Artin symbol
• Gauss's lemma