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.
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, 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) 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 .
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
whenever and are coprime.
- ^ 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
- 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