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

$\mathrm{N}\mathfrak{𝔭}:=|{{𝒪}}_k/\mathfrak{𝔭}|.$

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

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

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

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

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 ${{𝒪}}_k,$ and is denoted by

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

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):

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

In all cases (zero and nonzero)

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

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

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

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

Relation to the Hilbert symbol

The n-th power residue symbol is related to the Hilbert symbol $(\cdot ,\cdot {)}_{\mathfrak{𝔭}}$ 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 $K_{\mathfrak{𝔭}}$.

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

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

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

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

See also

• Modular_arithmetic#Residue_class
• Quadratic_residue#Prime_power_modulus
• 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, 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, Springer Monographs in Mathematics, 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, vol. 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021