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.

See also

• Artin symbol
• Gauss's lemma

Notes

1. Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers.
2. All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2
3. Neukirch (1999) p. 336
4. 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, doi:10.1007/978-3-662-12893-0, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002
• 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