Power residue symbol

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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[edit]

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

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.

  • If then
  • If then is not an -th power modulo
  • If then may or may not be an -th power modulo

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]

Notes[edit]

  1. ^ Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers.
  2. ^ All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2
  3. ^ Neukirch (1999) p. 336
  4. ^ 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, 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