Power residue symbol

(Redirected from Power reciprocity law)

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

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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

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This root of unity is called the n-th power residue symbol for   and is denoted by

 

Properties

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The n-th power symbol has properties completely analogous to those of the classical (quadratic) Jacobi symbol (  is a fixed primitive  -th root of unity):

 

In all cases (zero and nonzero)

 
 
 

All power residue symbols mod n are Dirichlet characters mod n, and the m-th power residue symbol only contains the m-th roots of unity, the m-th power residue symbol mod n exists if and only if m divides   (the Carmichael lambda function of n).

Relation to the Hilbert symbol

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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

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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

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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

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Notes

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  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

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  • 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