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 $\mathcal_k$ that contains a primitive ''n''-th root of unity $\zeta_n.$

Let $\mathfrak \subset \mathcal_k$ be a prime ideal and assume that ''n'' and $\mathfrak$ are coprime (i.e. $n \not \in \mathfrak$.)

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

:$\mathrm \mathfrak := , \mathcal_k / \mathfrak, .$

An analogue of Fermat's theorem holds in $\mathcal_k.$ If $\alpha \in \mathcal_k - \mathfrak,$ then
:$\alpha^\equiv 1 \bmod.$

And finally, suppose $\mathrm \mathfrak \equiv 1 \bmod.$ These facts imply that

:$\alpha^\equiv \zeta_n^s\bmod$

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 $\mathcal_k,$ and is denoted by

:$\left(\frac\right)_n= \zeta_n^s \equiv \alpha^\bmod.$

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):

:$\left(\frac\right)_n = \begin 0 & \alpha\in\mathfrak\\ 1 & \alpha\not\in\mathfrak\text \exists \eta \in\mathcal_k : \alpha \equiv \eta^n \bmod\\ \zeta & \alpha\not\in\mathfrak\text\eta \end$

In all cases (zero and nonzero)

:$\left(\frac\right)_n \equiv \alpha^\bmod.$
:$\left(\frac\right)_n \left(\frac\right)_n = \left(\frac\right)_n$
:$\alpha \equiv\beta\bmod \quad \Rightarrow \quad \left(\frac\right)_n = \left(\frac\right)_n$

Relation to the Hilbert symbol

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

:$\left(\frac\right)_n = (\pi, \alpha)_$

in the case $\mathfrak$ coprime to ''n'', where $\pi$ is any uniformising element for the local field $K_$.Neukirch (1999) p. 336

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 $\mathfrak\subset\mathcal_k$ is the product of prime ideals, and in one way only:
:$\mathfrak = \mathfrak_1 \cdots\mathfrak_g.$

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

:$\left(\frac\right)_n = \left(\frac\right)_n \cdots \left(\frac\right)_n.$

For $0 \neq \beta\in\mathcal_k$ then we define
:$\left(\frac\right)_n := \left(\frac\right)_n,$

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 $\alpha\equiv\beta\bmod$ then $\left(\tfrac\right)_n = \left(\tfrac\right)_n.$
*$\left(\tfrac\right)_n \left(\tfrac\right)_n = \left(\tfrac\right)_n.$
* $\left(\tfrac\right)_n \left(\tfrac\right)_n = \left(\tfrac\right)_n.$

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

Power reciprocity law

The ''power reciprocity law'', the analogue of the law of quadratic reciprocity, may be formulated in terms of the Hilbert symbols asNeukirch (1999) p. 415

:$\left(\right)_n \left(\right)_n^ = \prod_ (\alpha,\beta)_,$

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

See also

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

Notes

References

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Algebraic number theory