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]
Table of Contents
Background and notation[edit]
Let k be an algebraic number field with ring of integers
that contains a primitive n-th root of unity
after-content-x4
Let
be a prime ideal and assume that n and
are coprime (i.e.
.)
The norm of
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
If
then
And finally, suppose
These facts imply that
is well-defined and congruent to a unique
-th root of unity
Definition[edit]
This root of unity is called the n-th power residue symbol for
and is denoted by
Properties[edit]
The n-th power symbol has properties completely analogous to those of the classical (quadratic) Legendre symbol (
is a fixed primitive
-th root of unity):
In all cases (zero and nonzero)
Relation to the Hilbert symbol[edit]
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
.[3]
Generalizations[edit]
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.
Power reciprocity law[edit]
The power reciprocity law, the analogue of the law of quadratic reciprocity, may be formulated in terms of the Hilbert symbols as[4]
whenever
and
are coprime.
See also[edit]
^Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers.
^All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2
^Neukirch (1999) p. 336
^Neukirch (1999) p. 415
References[edit]
Gras, Georges (2003), Class field theory. From theory to practice, Springer Monographs in Mathematics, Berlin: Springer-Verlag, pp. 204–207, ISBN 3-540-44133-6, Zbl 1019.11032
Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer Science+Business Media, ISBN 0-387-97329-X
Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Springer Monographs in Mathematics, Berlin: Springer Science+Business Media, doi:10.1007/978-3-662-12893-0, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002
Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021
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