Power residue symbol explained

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

l{O} _{k} that contains a primitive n-th root of unity \zeta _{n.}

Let

ak{p} \subset l{O} _{k} be a prime ideal and assume that n and ak{p} are coprime (i.e. n \not \in ak{p} .)

The norm of

ak{p} is defined as the cardinality of the residue class ring (note that since ak{p} is prime the residue class ring is a finite field):N ak{p} : = | l{O} _{k} / ak{p}|.

An analogue of Fermat's theorem holds in

l{O} _{k.} If \alpha \in l{O} _{k} - ak{p}, then \alpha ^{N} -1 }\equiv 1 \bmod{ ak{p}}.

And finally, suppose

N ak{p} \equiv 1 \bmod{n}. These facts imply that\alpha ^{(N ak{p)/(-1)} {n}}\equiv \zeta ^{s\bmod{ ak{p} }} _{n}

is well-defined and congruent to a unique

n -th root of unity \zeta ^{s.} _{n}

Definition

This root of unity is called the n-th power residue symbol for

l{O} _{k,} and is denoted by\left( (\alpha)/(ak{p) }\right) _{n=} \zeta ^{s} _{n} \equiv \alpha ^{(N ak{p)/(-1)} {n}}\bmod{ ak{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( (\alpha)/(ak{p) }\right) _{n} = \begin{cases} 0 & \alpha\in ak{p}\\ 1 & \alpha\not\in ak{p} and \exists η \in l{O} _{k} : \alpha \equiv η ^{n} \bmod{ ak{p}}\\ \zeta & \alpha\not\in ak{p} andthereisnosuch η \end{cases}

In all cases (zero and nonzero)

\left( (\alpha)/(ak{p) }\right)_n \equiv \alpha^\bmod. \left( (\alpha)/(ak{p) }\right)_n \left(\frac\right)_n = \left(\frac\right)_n \alpha \equiv\beta\bmod{ ak{p}} ⇒ \left( (\alpha)/(ak{p) }\right) _{n} = \left( (\beta)/(ak{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

λ (n) (the Carmichael lambda function of n).

Relation to the Hilbert symbol

( ⋅ , ⋅ ) _{ak{p} } for the prime ak{p} by\left( (\alpha)/(ak{p) }\right) _{n} = (\pi, \alpha) _{ak{p} }

in the case

ak{p} coprime to n, where \pi is any uniformising element for the local field K _{ak{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

ak{a}\subset l{O} _{k} is the product of prime ideals, and in one way only:ak{a} = ak{p} _{1} … ak{p} _{g.}

The

n -th power symbol is extended multiplicatively:\left( (\alpha)/(ak{a) }\right) _{n} = \left( (\alpha)/(ak{p) _{1} }\right) _{n} … \left( (\alpha)/(ak{p) _{g} }\right) _{n.}

For

0 ≠ \beta\in l{O} _{k} then we define\left( (\alpha)/(\beta) \right) _{n} : = \left( (\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{ ak{a}} then \left(\tfrac{\alpha}{ ak{a} }\right) _{n} = \left(\tfrac{\beta}{ ak{a} }\right) _{n.} \left(\tfrac{\alpha}{ ak{a} }\right) _{n} \left(\tfrac{\beta}{ ak{a} }\right) _{n} = \left(\tfrac{\alpha\beta}{ ak{a} }\right) _{n.} \left(\tfrac{\alpha}{ ak{a} }\right) _{n} \left(\tfrac{\alpha}{ ak{b} }\right) _{n} = \left(\tfrac{\alpha}{ ak{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 η ^{n\bmod{ak{a}}} then \left(\tfrac{\alpha}{ ak{a} }\right) _{n} =1 .

• If

\left(\tfrac{\alpha}{ ak{a} }\right) _{n} ≠ 1 then \alpha is not an n -th power modulo ak{a}.

• If

\left(\tfrac{\alpha}{ ak{a} }\right) _{n} =1 then \alpha may or may not be an n -th power modulo ak{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({ (\alpha)/(\beta) }\right)_n \left(\right)_n^ = \prod_ (\alpha,\beta)_,

whenever

\alpha and \beta are coprime.

See also

• Modular_arithmetic#Residue_class
• Quadratic_residue#Prime_power_modulus
• Artin symbol
• Gauss's lemma

Notes and References

1. [Quadratic reciprocity]
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