Power residue symbol
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]
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 (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, 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 ( 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 for the prime by
in the case coprime to n, where is any uniformising element for the local field .[3]
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[4]
whenever and are coprime.
See also
Notes
- ↑ 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
- 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, 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, 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021