In
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
the ''n''-th power residue symbol (for an integer ''n'' > 2) is a generalization of the (quadratic)
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...
to ''n''-th powers. These symbols are used in the statement and proof of
cubic,
quartic,
Eisenstein, and related higher
reciprocity law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irr ...
s.
Background and notation
Let ''k'' be an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
with
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
that contains a
primitive ''n''-th root of unity
Let
be a
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
and assume that ''n'' and
are
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
(i.e.
.)
The
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
of
is defined as the cardinality of the residue class ring (note that since
is prime the residue class ring is a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
):
:
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 (quadratic)
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...
(
is a fixed primitive
-th root of unity):
:
In all cases (zero and nonzero)
:
:
:
Relation to the Hilbert symbol
The ''n''-th power residue symbol is related to the
Hilbert symbol In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity l ...
for the prime
by
:
in the case
coprime to ''n'', where
is any
uniformising element for the
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
.
[Neukirch (1999) p. 336]
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
Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a ...
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
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
, may be formulated in terms of the
Hilbert symbol In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity l ...
s as
[Neukirch (1999) p. 415]
:
whenever
and
are coprime.
See also
*
Modular_arithmetic#Residue_class
*
Quadratic_residue#Prime_power_modulus
*
Artin symbol The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line ...
*
Gauss's lemma
Notes
References
*
*
*
*
Algebraic number theory