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.
Contents
Background and notation
Let k be an algebraic number field with ring of integers
Let
The norm of
An analogue of Fermat's theorem holds in
And finally,
is well-defined and congruent to a unique
Definition
This root of unity is called the n-th power residue symbol for
Properties
The n-th power symbol has properties completely analogous to those of the classical (quadratic) Legendre symbol (
In all cases (zero and nonzero)
Relation to the Hilbert symbol
The n-th power residue symbol is related to the Hilbert symbol
in the case
Generalizations
The
Any ideal
The
For
where
Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.
Since the symbol is always an
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