Cubic reciprocity

Cubic reciprocity is a collection of theorems in elementary and algebraic number theory. These theorems describe when a certain kind of congruence can be solved. The congruence is written as x³ ≡ p (mod q).

The word "reciprocity" comes from the form of the main theorem. It says that if p and q are primary numbers in the ring of Eisenstein integers, and both are coprime to 3, then x³ ≡ p (mod q) can be solved if and only if x³ ≡ q (mod p) can be solved. In other words, the two congruences are linked, so each one tells us something about the other.

History

Sometime before 1748, Euler made the first guesses about which small whole numbers are cubic residues, but these were not published until 1849, 62 years after his death. Gauss mentioned cubic residues and reciprocity in three published works. His Disquisitiones Arithmeticae (1801) had one result about cubic residues. In 1818, he explained that he was publishing his fifth and sixth proofs of quadratic reciprocity because their methods, Gauss's lemma and Gaussian sums, could be used for cubic and biquadratic reciprocity. A footnote in his 1832 work on biquadratic reciprocity said that cubic reciprocity is easiest to describe using the Eisenstein integers.

Gauss's diary and other unpublished notes suggest he knew the rules for cubic and quartic residues by 1805, and that he found the full theorems and proofs of cubic and biquadratic reciprocity around 1814. These proofs were found in his papers after his death, but it is not clear whether they were his or Eisenstein's. Jacobi published several theorems about cubic residues in 1827 without proofs, but he presented proofs in his lectures at Königsberg in 1836–37. The first published proofs were by Eisenstein in 1844.

A cubic residue modulo p is a number that matches the third power of a whole number when divided by p. If no whole number's cube matches it, the number is a cubic nonresidue. Cubic residues are usually only studied for moduli where the Carmichael lambda function of n is divisible by 3, because for other numbers every residue is a cubic residue. Mathematicians often work with prime numbers. If q ≡ 2 (mod 3) is prime, every number is a cubic residue modulo q, as shown by Fermat's little theorem. So the interesting case is when p ≡ 1 (mod 3). Then the nonzero residues split into three equal groups. In the language of group theory, the cubic residues form a subgroup of index 3, and the other two groups are its cosets. A primitive root can also be used to describe these groups.

Fermat showed that every prime p ≡ 1 (mod 3) can be written as a² + 3b², and this way of writing it is unique, except for the signs of a and b. Using this form, Euler's conjectures, which are now proved, describe when small numbers such as 2, 3, 5, 6 and 7 are cubic residues of such a prime. Gauss's theorem gives a related result about these primes. Jacobi's theorem is stated without proof, and Lehmer's theorem gives another way to test when one prime is a cubic residue of another. Martinet's theorem connects pairs of primes, and Sharifi's theorem says that when p = 1 + 3x + 9x² is prime, any divisor of x is a cubic residue modulo p.

Eisenstein integers

Gauss said that the theorems on biquadratic residues are simplest when the number system is extended to imaginary numbers of the form a + bi. These are now called the Gaussian integers. He also said that cubic residues need a similar idea, using imaginary numbers built from a root of unity. In his first work on cubic reciprocity, Eisenstein developed the numbers built from a cube root of unity, now called the Eisenstein integers. Both systems are unique factorization domains. The other imaginary numbers needed for higher powers are the rings of integers of cyclotomic number fields, and the Gaussian and Eisenstein integers are the simplest examples. The Eisenstein integers, written as a + bω, form a Euclidean domain with a norm function, and their group of units is a cyclic group of sixth roots of unity. Each prime falls into one of three groups: the prime 3 ramifies, primes congruent to 2 mod 3 remain inert, and primes congruent to 1 mod 3 split. An analogue of Fermat's little theorem holds in this system, and it allows a "cubic residue character" to be defined, which behaves a bit like the Legendre symbol. Working with these numbers uses ideas such as congruence, greatest common divisor, complex conjugation, and the multiplicative inverse of a number. The cubic character can also be extended to composite numbers in a way similar to the Jacobi symbol.

The main result, called the cubic reciprocity theorem, says that for two primary numbers α and β, the cubic residue character of α modulo β equals that of β modulo α. Supplementary theorems cover the units and the prime 1 − ω, showing how their cubic characters depend on the parts of a primary number.