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] Let k be an algebraic number field with ring of integers
that contains a primitive n-th root of unity
be a prime ideal and assume that n and
is defined as the cardinality of the residue class ring (note that since
is prime the residue class ring is a finite field): An analogue of Fermat's theorem holds in
then And finally, suppose
These facts imply that is well-defined and congruent to a unique
-th root of unity
This root of unity is called the n-th power residue symbol for
and is denoted by The n-th power symbol has properties completely analogous to those of the classical (quadratic) Jacobi symbol (
is a fixed primitive
-th root of unity): In all cases (zero and nonzero) 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
(the Carmichael lambda function of n).
The n-th power residue symbol is related to the Hilbert symbol
is any uniformising element for the local field
-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.
is the product of prime ideals, and in one way only: The
-th power symbol is extended multiplicatively: For
is the principal ideal generated by
Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.
-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.
The power reciprocity law, the analogue of the law of quadratic reciprocity, may be formulated in terms of the Hilbert symbols as[4] whenever