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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

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

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Definition

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

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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).

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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]

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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
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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.

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See also

Notes

References

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