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 [math]\displaystyle{ \mathcal{O}_k }[/math] that contains a primitive n-th root of unity [math]\displaystyle{ \zeta_n. }[/math]

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

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

[math]\displaystyle{ \mathrm{N} \mathfrak{p} := |\mathcal{O}_k / \mathfrak{p}|. }[/math]

An analogue of Fermat's theorem holds in [math]\displaystyle{ \mathcal{O}_k. }[/math] If [math]\displaystyle{ \alpha \in \mathcal{O}_k - \mathfrak{p}, }[/math] then

[math]\displaystyle{ \alpha^{\mathrm{N} \mathfrak{p} -1}\equiv 1 \bmod{\mathfrak{p}}. }[/math]

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

[math]\displaystyle{ \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\equiv \zeta_n^s\bmod{\mathfrak{p} } }[/math]

is well-defined and congruent to a unique [math]\displaystyle{ n }[/math]-th root of unity [math]\displaystyle{ \zeta_n^s. }[/math]

Definition

This root of unity is called the n-th power residue symbol for [math]\displaystyle{ \mathcal{O}_k, }[/math] and is denoted by

[math]\displaystyle{ \left(\frac{\alpha}{\mathfrak{p} }\right)_n= \zeta_n^s \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\bmod{\mathfrak{p}}. }[/math]

Properties

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

[math]\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} }[/math]

In all cases (zero and nonzero)

[math]\displaystyle{ \left(\frac{\alpha}{\mathfrak{p}}\right)_n \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\bmod{\mathfrak{p}}. }[/math]

[math]\displaystyle{ \left(\frac{\alpha}{\mathfrak{p}}\right)_n \left(\frac{\beta}{\mathfrak{p}}\right)_n = \left(\frac{\alpha\beta}{\mathfrak{p} }\right)_n }[/math]

[math]\displaystyle{ \alpha \equiv\beta\bmod{\mathfrak{p}} \quad \Rightarrow \quad \left(\frac{\alpha}{\mathfrak{p} }\right)_n = \left(\frac{\beta}{\mathfrak{p} }\right)_n }[/math]

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 [math]\displaystyle{ \lambda(n) }[/math] (the Carmichael lambda function of n).

Relation to the Hilbert symbol

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

[math]\displaystyle{ \left(\frac{\alpha}{\mathfrak{p} }\right)_n = (\pi, \alpha)_{\mathfrak{p}} }[/math]

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

Generalizations

The [math]\displaystyle{ n }[/math]-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 [math]\displaystyle{ \mathfrak{a}\subset\mathcal{O}_k }[/math] is the product of prime ideals, and in one way only:

[math]\displaystyle{ \mathfrak{a} = \mathfrak{p}_1 \cdots\mathfrak{p}_g. }[/math]

The [math]\displaystyle{ n }[/math]-th power symbol is extended multiplicatively:

[math]\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. }[/math]

For [math]\displaystyle{ 0 \neq \beta\in\mathcal{O}_k }[/math] then we define

[math]\displaystyle{ \left(\frac{\alpha}{\beta}\right)_n := \left(\frac{\alpha}{(\beta) }\right)_n, }[/math]

where [math]\displaystyle{ (\beta) }[/math] is the principal ideal generated by [math]\displaystyle{ \beta. }[/math]

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

• If [math]\displaystyle{ \alpha\equiv\beta\bmod{\mathfrak{a}} }[/math] then [math]\displaystyle{ \left(\tfrac{\alpha}{\mathfrak{a} }\right)_n = \left(\tfrac{\beta}{\mathfrak{a} }\right)_n. }[/math]
• [math]\displaystyle{ \left(\tfrac{\alpha}{\mathfrak{a} }\right)_n \left(\tfrac{\beta}{\mathfrak{a} }\right)_n = \left(\tfrac{\alpha\beta}{\mathfrak{a} }\right)_n. }[/math]
• [math]\displaystyle{ \left(\tfrac{\alpha}{\mathfrak{a} }\right)_n \left(\tfrac{\alpha}{\mathfrak{b} }\right)_n = \left(\tfrac{\alpha}{\mathfrak{ab} }\right)_n. }[/math]

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

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

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

[math]\displaystyle{ \left({\frac{\alpha}{\beta}}\right)_n \left({\frac{\beta}{\alpha}}\right)_n^{-1} = \prod_{\mathfrak{p} | n\infty} (\alpha,\beta)_{\mathfrak{p}}, }[/math]

whenever [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] are coprime.

See also

• Modular arithmetic
• Quadratic residue
• Artin symbol
• Gauss's lemma

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 
• 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 
• 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