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.

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

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

  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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