Girish Mahajan (Editor)

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 reciprocity laws.

Contents

Background and notation

Let k be an algebraic number field with ring of integers O k that contains a primitive n-th root of unity ζ n .

Let p O k be a prime ideal and assume that n and p are coprime (i.e. n p .)

The norm of p is defined as the cardinality of the residue class ring (note that since p is prime the residue class ring is a finite field):

N p := | O k / p | .

An analogue of Fermat's theorem holds in O k . If α O k p , then

α N p 1 1 mod p .

And finally, N p 1 mod n . These facts imply that

α N p 1 n ζ n s mod p

is well-defined and congruent to a unique n -th root of unity ζ n s .

Definition

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

( α p ) n = ζ n s α N p 1 n mod p .

Properties

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

( α p ) n = { 0 α p 1 α p  and  η O k : α η n mod p ζ α p  and there is no such  η

In all cases (zero and nonzero)

( α p ) n α N p 1 n mod p . ( α p ) n ( β p ) n = ( α β p ) n α β mod p ( α p ) n = ( β p ) n

Relation to the Hilbert symbol

The n-th power residue symbol is related to the Hilbert symbol ( , ) p for the prime p by

( α p ) n = ( π , α ) p

in the case p coprime to n, where π is any uniformising element for the local field K p .

Generalizations

The n -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 a O k is the product of prime ideals, and in one way only:

a = p 1 p g .

The n -th power symbol is extended multiplicatively:

( α a ) n = ( α p 1 ) n ( α p g ) n .

For 0 β O k then we define

( α β ) n := ( α ( β ) ) n ,

where ( β ) is the principal ideal generated by β .

Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters.

  • If α β mod a then ( α a ) n = ( β a ) n .
  • ( α a ) n ( β a ) n = ( α β a ) n .
  • ( α a ) n ( α b ) n = ( α a b ) n .
  • Since the symbol is always an n -th root of unity, because of its multiplicativity it is equal to 1 whenever one parameter is an n -th power; the converse is not true.

  • If α η n mod a then ( α a ) n = 1.
  • If ( α a ) n 1 then α is not an n -th power modulo a .
  • If ( α a ) n = 1 then α may or may not be an n -th power modulo a .
  • 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

    ( α β ) n ( β α ) n 1 = p | n ( α , β ) p ,

    whenever α and β are coprime.

    References

    Power residue symbol Wikipedia