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 reciprocity laws.

Background and notation

Let k be an algebraic number field with ring of integers ${𝒪}_k$ that contains a primitive n-th root of unity ${\zeta }_n.$

Let $𝔭\subset {𝒪}_k$ be a prime ideal and assume that n and $𝔭$ are coprime (i.e. $n\in ̸𝔭$.)

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

$\mathrm{N}𝔭:=|{𝒪}_k/𝔭|.$

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

${\alpha }^{\mathrm{N}𝔭-1}\equiv 1mod𝔭.$

And finally, suppose $\mathrm{N}𝔭\equiv 1modn.$ These facts imply that

${\alpha }^{\frac{\mathrm{N}𝔭-1}{n}}\equiv {\zeta }_n^{s}mod𝔭$

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 ${𝒪}_k,$ and is denoted by

${\left(\frac{\alpha }{𝔭}\right)}_n={\zeta }_n^s\equiv {\alpha }^{\frac{\mathrm{N}𝔭-1}{n}}mod𝔭.$

Properties

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

${\left(\frac{\alpha }{𝔭}\right)}_n=\{\begin{array}{cc}0& \alpha \in 𝔭\\ 1& \alpha \in ̸𝔭\text{ and }\mathrm{\exists }\eta \in {𝒪}_k:\alpha \equiv {\eta }^{n}mod𝔭\\ \zeta & \alpha \in ̸𝔭\text{ and there is no such }\eta \end{array}$

In all cases (zero and nonzero)

${\left(\frac{\alpha }{𝔭}\right)}_n\equiv {\alpha }^{\frac{\mathrm{N}𝔭-1}{n}}mod𝔭.$

${\left(\frac{\alpha }{𝔭}\right)}_n{\left(\frac{\beta }{𝔭}\right)}_n={\left(\frac{\alpha \beta }{𝔭}\right)}_n$

$\alpha \equiv \beta mod𝔭\Rightarrow {\left(\frac{\alpha }{𝔭}\right)}_n={\left(\frac{\beta }{𝔭}\right)}_n$

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 $\lambda (n)$ (the Carmichael lambda function of n).

Relation to the Hilbert symbol

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

${\left(\frac{\alpha }{𝔭}\right)}_n=(\pi ,\alpha {)}_{𝔭}$

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

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 $𝔞\subset {𝒪}_k$ is the product of prime ideals, and in one way only:

$𝔞={𝔭}_1\cdots {𝔭}_g.$

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

${\left(\frac{\alpha }{𝔞}\right)}_n={\left(\frac{\alpha }{{𝔭}_1}\right)}_n\cdots {\left(\frac{\alpha }{{𝔭}_g}\right)}_n.$

For $0\ne \beta \in {𝒪}_k$ 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 $\alpha \equiv \beta mod𝔞$ then ${\left(\frac{\alpha }{𝔞}\right)}_n={\left(\frac{\beta }{𝔞}\right)}_n.$
• ${\left(\frac{\alpha }{𝔞}\right)}_n{\left(\frac{\beta }{𝔞}\right)}_n={\left(\frac{\alpha \beta }{𝔞}\right)}_n.$
• ${\left(\frac{\alpha }{𝔞}\right)}_n{\left(\frac{\alpha }{𝔟}\right)}_n={\left(\frac{\alpha }{𝔞𝔟}\right)}_n.$

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 $\alpha \equiv {\eta }^{n}mod𝔞$ then ${\left(\frac{\alpha }{𝔞}\right)}_n=1.$
• If ${\left(\frac{\alpha }{𝔞}\right)}_n\ne 1$ then $\alpha$ is not an $n$-th power modulo $𝔞.$
• If ${\left(\frac{\alpha }{𝔞}\right)}_n=1$ then $\alpha$ may or may not be an $n$-th power modulo $𝔞.$

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

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

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

See also

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

References

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