Power residue symbol

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.

Notes

References

''n''^th 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 Then letting t=s-r,;;\zeta_n^t\equiv 1 \bmod\mathfrak{p}, and 0 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 − 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 ζn 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 ζn**s.

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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References

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
5. Lemmermeyer, Ch. 4.1
6. Lemmermeyer, Prop 4.1