Power residue symbol


In algebraic number theory the n-th power residue symbol is a generalization of the 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 that contains a primitive n-th root of unity
Let be a prime ideal and assume that n and are coprime
The norm of is defined as the cardinality of the residue class ring :
An analogue of Fermat's theorem holds in If then
And finally, suppose These facts imply that
is well-defined and congruent to a unique -th root of unity

Definition

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

Properties

The n-th power symbol has properties completely analogous to those of the classical Jacobi symbol :
In all cases
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 .

Relation to the Hilbert symbol

The n-th power residue symbol is related to the Hilbert symbol for the prime by
in the case coprime to n, where is any uniformising element for the local field.

Generalizations

The -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 is the product of prime ideals, and in one way only:
The -th power symbol is extended multiplicatively:
For then we define
where is the principal ideal generated by
Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.
  • If then
Since the symbol is always an -th root of unity, because of its multiplicativity it is equal to 1 whenever one parameter is an -th power; the converse is not true.
  • If then
  • If then is not an -th power modulo
  • If then may or may not be an -th power modulo

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
whenever and are coprime.