Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x3 ≡ p (mod q) is solvable if and only if x3 ≡ q (mod p) is solvable.
History
Sometime before 1748 Euler made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, 62 years after his death.
Gauss's published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in the Disquisitiones Arithmeticae (1801). In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818) he said that he was publishing these proofs because their techniques (Gauss's lemma and Gaussian sums, respectively) can be applied to cubic and biquadratic reciprocity. Finally, a footnote in the second (of two) monographs on biquadratic reciprocity (1832) states that cubic reciprocity is most easily described in the ring of Eisenstein integers.
From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of cubic and biquadratic reciprocity around 1814. Proofs of these were found in his posthumous papers, but it is not clear if they are his or Eisenstein's.
Jacobi published several theorems about cubic residuacity in 1827, but no proofs. In his Königsberg lectures of 1836–37 Jacobi presented proofs. The first published proofs were by Eisenstein (1844).
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 easier to work modulo prime numbers, so in this section all moduli p, q, etc., are assumed to be positive odd primes.
We first note that if q ≡ 2 (mod 3) is a prime then every number is a cubic residue modulo 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,
Multiplying the two congruences we have
Now substituting 3n + 2 for q we have:
Therefore, the only interesting case is when the modulus p ≡ 1 (mod 3). In this case the non-zero residue classes (mod p) can be divided into three sets, each containing (p−1)/3 numbers. Let e be a cubic non-residue. 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 (resp. second, third) set is the numbers whose indices with respect to this root are congruent to 0 (resp. 1, 2) (mod 3). In the vocabulary of group theory, the cubic residues form a subgroup of index 3 of the multiplicative group and the three sets are its cosets.
Primes ≡ 1 (mod 3)
A theorem of Fermat states that every prime p ≡ 1 (mod 3) can be written as p = a2 + 3b2 and (except for the signs of a and b) this representation is unique.
Letting m = a + b and n = a − b, we see that this is equivalent to p = m2 − mn + n2 (which equals (n − m)2 − (n − m)n + n2 = m2 + m(n − m) + (n − m)2, so m and n are not determined uniquely). Thus,
and it is a straightforward exercise to show that exactly one of m, n, or m − n is a multiple of 3, so
and this representation is unique up to the signs of L and M.
For relatively prime integers m and n define the rational cubic residue symbol as
It is important to note that this symbol does not have the multiplicative properties of the Legendre symbol; for this, we need the true cubic character defined below.
Euler's Conjectures. Let p = a2 + 3b2 be a prime. Then the following hold:
The first two can be restated as follows. Let p be a prime that is congruent to 1 modulo 3. Then: 2 is a cubic residue of p if and only if p = a2 + 27b2. 3 is a cubic residue of p if and only if 4p = a2 + 243b2.
Gauss's Theorem. Let p be a positive prime such that
Then
One can easily see that Gauss's Theorem implies:
Jacobi's Theorem (stated without proof). Let q ≡ p ≡ 1 (mod 6) be positive primes. Obviously both p and q are also congruent to 1 modulo 3, therefore assume:
Let x be a solution of x2 ≡ −3 (mod q). Then
and we have:
Lehmer's Theorem. Let q and p be primes, with Then: