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.
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.
[2] In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818)[3] 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.
[4] From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of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.
[5][6] Proofs of these were found in his posthumous papers, but it is not clear if they are his or Eisenstein's.
[7] Jacobi published several theorems about cubic residuacity in 1827, but no proofs.
[12] We first note that if q ≡ 2 (mod 3) is a prime then every number is a cubic residue modulo 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.
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.
A theorem of Fermat[13][14] 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.[15] 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.
The first few examples[26] of this are equivalent to Euler's conjectures: Since obviously L ≡ M (mod 2), the criterion for q = 2 can be simplified as: In his second monograph on biquadratic reciprocity, Gauss says: The theorems on biquadratic residues gleam with the greatest simplicity and genuine beauty only when the field of arithmetic is extended to imaginary numbers, so that without restriction, the numbers of the form a + bi constitute the object of study ... we call such numbers integral complex numbers.
[29] [bold in the original]These numbers are now called the ring of Gaussian integers, denoted by Z[i].
In a footnote he adds The theory of cubic residues must be based in a similar way on a consideration of numbers of the form a + bh where h is an imaginary root of the equation h3 = 1 ... and similarly the theory of residues of higher powers leads to the introduction of other imaginary quantities.
[30]In his first monograph on cubic reciprocity[31] Eisenstein developed the theory of the numbers built up from a cube root of unity; they are now called the ring of Eisenstein integers.
Eisenstein said that to investigate the properties of this ring one need only consult Gauss's work on Z[i] and modify the proofs.
The "other imaginary quantities" needed for the "theory of residues of higher powers" are the rings of integers of the cyclotomic number fields; the Gaussian and Eisenstein integers are the simplest examples of these.
(the elements with a multiplicative inverse or equivalently those with unit norm) is a cyclic group of the sixth roots of unity,
The primes fall into three classes:[32] A number is primary if it is coprime to 3 and congruent to an ordinary integer modulo
The notions of congruence[33] and greatest common divisor[34] are defined the same way in
and is denoted by[36] The cubic residue character has formal properties similar to those of the Legendre symbol: Let α and β be primary.
Then There are supplementary theorems[40][41] for the units and the prime 1 − ω: Let α = a + bω be primary, a = 3m + 1 and b = 3n.
(If a ≡ 2 (mod 3) replace α with its associate −α; this will not change the value of the cubic characters.)
The references to the original papers of Euler, Jacobi, and Eisenstein were copied from the bibliographies in Lemmermeyer and Cox, and were not used in the preparation of this article.
182–283 of The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76.
Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art.
47–64 German translations of all three of the above are the following, which also has the Disquisitiones Arithmeticae and Gauss's other papers on number theory.