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 $$\mathcal{O}_k$$ 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, suppose $$\mathrm{N} \mathfrak{p} \equiv 1 \bmod{n}.$$ 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) Jacobi 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)


 * $$\left(\frac{\alpha}{\mathfrak{p}}\right)_n \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\bmod{\mathfrak{p}}. $$
 * $$ \left(\frac{\alpha}{\mathfrak{p}}\right)_n \left(\frac{\beta}{\mathfrak{p}}\right)_n = \left(\frac{\alpha\beta}{\mathfrak{p} }\right)_n $$
 * $$\alpha \equiv\beta\bmod{\mathfrak{p}} \quad \Rightarrow \quad \left(\frac{\alpha}{\mathfrak{p} }\right)_n  = \left(\frac{\beta}{\mathfrak{p} }\right)_n  $$

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 $$\mathfrak{a}\subset\mathcal{O}_k$$ 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


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

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

nth power residue symbol
Let k be an algebraic number field with ring of integers  $$\mathcal{O}_k,$$   and let   $$\mathfrak{p} \subset \mathcal{O}_k $$   be a prime ideal. The norm of $$\mathfrak{p} $$ is defined as the cardinality of the residue class ring (since $$\mathfrak{p} $$ is prime this is a finite field)   $$ \mathcal{O}_k / \mathfrak{p}\;:\;\;\; \mathrm{N} \mathfrak{p} = |\mathcal{O}_k / \mathfrak{p}|.$$

Assume that a primitive nth root of unity  $$\zeta_n\in\mathcal{O}_k,$$   and that n and $$\mathfrak{p} $$ are coprime (i.e.$$n\not\in \mathfrak{p}.$$)  Then

No two distinct nth roots of unity can be congruent $$\bmod\mathfrak{p}.$$

The proof is by contradiction: assume otherwise, that   $$\zeta_n^r\equiv\zeta_n^s\bmod\mathfrak{p}, \;\;0 <r<s\le n.$$  Then letting  $$t=s-r,\;\;\zeta_n^t\equiv 1 \bmod\mathfrak{p}, $$  and $$ 0 < t< n.\  $$ From the definition of roots of unity,
 * $$x^n-1=(x-1)(x-\zeta_n)(x-\zeta_n^2)\dots(x-\zeta_n^{n-1}),$$  and dividing by  x &minus; 1  gives
 * $$x^{n-1}+x^{n-2}+\dots +x + 1 =(x-\zeta_n)(x-\zeta_n^2)\dots(x-\zeta_n^{n-1}).$$

Letting x = 1 and taking residues $$\bmod\mathfrak{p},$$
 * $$n\equiv(1-\zeta_n)(1-\zeta_n^2)\dots(1-\zeta_n^{n-1})\bmod\mathfrak{p}.$$

Since n and $$ \mathfrak{p}$$ are coprime,$$ n\not\equiv 0\bmod\mathfrak{p},$$   but under the assumption, one of the factors on the right must be zero. Therefore the assumption that two distinct roots are congruent is false.

Thus the residue classes of  $$ \mathcal{O}_k / \mathfrak{p}$$   containing the powers of &zeta;n are a subgroup of order n of its (multiplicative) group of units,   $$(\mathcal{O}_k/\mathfrak{p}) ^\times  = \mathcal{O}_k /\mathfrak{p}- \{0\}.$$   Therefore the order of    $$(\mathcal{O}_k/\mathfrak{p})^ \times$$   is a multiple of n, and
 * $$\mathrm{N} \mathfrak{p} = |\mathcal{O}_k / \mathfrak{p}| = |(\mathcal{O}_k / \mathfrak{p} )^\times| + 1 \equiv 1 \bmod{n}.$$

There is an analogue of Fermat's theorem in $$\mathcal{O}_k:$$ If $$\alpha \in \mathcal{O}_k,\;\;\; \alpha\not\in \mathfrak{p},$$  then
 * $$\alpha^{\mathrm{N} \mathfrak{p} -1}\equiv 1 \bmod{\mathfrak{p} },

$$  and since   $$\mathrm{N} \mathfrak{p} \equiv 1 \bmod{n},$$


 * $$\alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\equiv \zeta_n^s\bmod{\mathfrak{p} }

$$ is well-defined and congruent to a unique nth root of unity &zeta;ns.

This root of unity is called the nth-power residue symbol for   $$\mathcal{O}_k,$$   and is denoted by


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

It can be proven that



\left(\frac{\alpha}{\mathfrak{p} }\right)_n= 1 \text{ if and only if there is an } \eta \in\mathcal{O}_k\;\;\text{ such that } \;\;\alpha\equiv\eta^n\bmod{\mathfrak{p}}.$$

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