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, octic, Artin, Eisenstein, and related higherQuadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers. reciprocity laws.All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2

Definition

a is called as nth power residue mod m when the equation has solution. We usually require a coprime to m, and sometimes we only consider n dividing , where is the Carmichael lambda function.

If we require a coprime to m and only consider n dividing (where is the Carmichael lambda function), then for n not divisible by 8, then a number a that are nth power residue mod every m must be an nth power, however, if n is divisible by 8, then a number a that is nth power residue mod every m need not be an nth power, the counterexamples are (16k8)r is (8r)th power residue mod every m, but not a (8r)th power. Also, if either n = 2 or n is odd, a number a that are nth power residue mod every prime p must be an nth power, however, if n is even and n > 2, then a number a that is nth power residue mod every prime p need not be an nth power, the counterexamples are: For n = 2qr with prime q (in fact, this is true for all squarefree q > 1), (qqk2q)r is (2qr)th power residue mod every prime p but not a (2qr)th power, if , (−qqk2q)r is (2qr)th power residue mod every prime p but not a (2qr)th power, if .

Names of nth power residues

n = 2: Quadratic residue n = 3: Cubic residue n = 4: Quartic residue (or Biquadratic residue) n = 5: Quintic residue n = 6: Sextic residue n = 7: Septic residue n = 8: Octic residue n = 9: Nonic residue n = 10: Decic residue

Theorems of power residues

m is rth power residue mod n if and only if (m mod n) is rth power residue mod n (i.e. whether m is rth power residue mod n is only dependent to m mod n) If m is rth power residue mod n and s divides r, then m is sth power residue mod n If m is nth power residue mod r and s divides r, then m is nth power residue mod s If m is both "rth power residue mod n" and "sth power residue mod n", and r and s are coprime, then m is (r×s)th power residue mod n (more generally, if m is both "rth power residue mod n" and "sth power residue mod n", then m is (lcm(r,s))th power residue mod n) If m is both "nth power residue mod r" and "nth power residue mod s", and r and s are coprime, then m is nth power residue mod r×s (more generally, if m is both "nth power residue mod r" and "nth power residue mod s", then m is nth power residue mod lcm(r,s)) If m is coprime to n and m is (gcd(r,λ(n)))th power residue mod n (where λ is the Carmichael function), then m is rth power residue mod n If m is (s×gcd(r,λ(n)))th power residue mod n and s×gcd(r,λ(n)) ≥ the maximum exponent in the prime factorization of n (), then m is rth power residue mod n If p is prime and r is coprime to p−1, then all integers m not divisible by p are rth power residue mod p (more generally, if r is coprime to λ(n) (where λ is the Carmichael function), then all integers m coprime to n are rth power residue mod n) If p is prime and r is not divisible by p and m is not divisible by p and m is rth power residue mod p, then m is rth power residue mod all powers of p (more generally, if r is coprime to n and m is coprime to n and m is rth power residue mod n, then m is rth power residue mod all numbers whose all prime factors are prime factors of n) If p is prime and m is not divisible by p and m is prth power residue mod ps and s ≥ r+1, then m is prth power residue mod all powers of p

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

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) 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 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 .Neukirch (1999) p. 336

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.

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 The power reciprocity law, the analogue of the law of quadratic reciprocity, may be formulated in terms of the Hilbert symbols asNeukirch (1999) p. 415

whenever and are coprime.

Power residue symbol

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