Cubic Reciprocity - Integers

Integers

A cubic residue (mod p) is any number congruent to the third power of an integer (mod p). If x3 ≡ a (mod p) does not have an integer solution, a is a cubic nonresidue (mod p).

As is often the case in number theory, it is easiest to work modulo prime numbers, so in this section all moduli p, q, etc., are assumed to be positive, odd primes.

The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 2 (mod 3) every number is a cubic residue (mod q). Let q = 3n + 2; since 0 = 03 is obviously a cubic residue, assume x is not divisible by q. Then by Fermat's little theorem,

is a cubic residue (mod q).

Therefore, the only interesting case is when the modulus p ≡ 1 (mod 3).

In this case, p ≡ 1 (mod 3), the nonzero residue classes (mod p) can be divided into three sets, each containing (p−1)/3 numbers. Let e be a cubic nonresidue. The first set is the cubic residues; the second one is e times the numbers in the first set, and the third is e2 times the numbers in the first set. Another way to describe this division is to let e be a primitive root (mod p); then the first (respectively second, third) set is the numbers whose indices with respect to this root are ≡ 0 (resp. 1, 2) (mod 3). In the vocabulary of group theory, the first set is a subgroup of index 3 (of the multiplicative group Z/pZ ×), and the other two are its cosets.