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, 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From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of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).
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 first set is a subgroup of index 3 of the multiplicative group and the other two are its cosets.
Primes ≡ 1 (mod 3)
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
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.
- 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.
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:
Note that the first condition implies: that any number that divides L or M is a cubic residue (mod p).
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. [bold in the original]
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.
In his first monograph on cubic reciprocity 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 (paraphrasing) "to investigate the properties of this ring one need only consult Gauss's work on Z[i] and modify the proofs". This is not surprising since both rings are unique factorization domains.
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.
Facts and terminology
And consider the ring of Eisenstein integers:
This is a Euclidean domain with the norm function given by:
Note that the norm is always congruent to 0 or 1 (mod 3).
The group of units in (the elements with a multiplicative inverse or equivalently those with unit norm) is a cyclic group of the sixth roots of unity,
is a unique factorization domain. The primes fall into three classes:
- 3 is a special case:
- It is the only prime in divisible by the square of a prime in . The prime 3 is said to ramify in .
- Positive primes in congruent to 2 (mod 3) are also primes in . These primes are said to remain inert in . Note that if is any inert prime then:
- Positive primes in congruent to 1 (mod 3) are the product of two conjugate primes in . These primes are said to split in . Their factorization is given by:
- for example
A number is primary if it is coprime to 3 and congruent to an ordinary integer modulo which is the same as saying it is congruent to modulo 3. If one of or is primary. Moreover, the product of two primary numbers is primary and the conjugate of a primary number is also primary.
The unique factorization theorem for is: if then
where each is a primary (under Eisenstein's definition) prime. And this representation is unique, up to the order of the factors.
The notions of congruence and greatest common divisor are defined the same way in as they are for the ordinary integers . Because the units divide all numbers, a congruence modulo is also true modulo any associate of , and any associate of a GCD is also a GCD.
Cubic residue character
An analogue of Fermat's little theorem is true in : if is not divisible by a prime ,
Now assume that so that Or put differently Then we can write:
The cubic residue character has formal properties similar to those of the Legendre symbol:
- If then
- where the bar denotes complex conjugation.
- If and are associates then
- The congruence has a solution in if and only if
- If are such that then
- The cubic character can be extended multiplicatively to composite numbers (coprime to 3) in the "denominator" in the same way the Legendre symbol is generalized into the Jacobi symbol. Like the Jacobi symbol, if the "denominator" of the cubic character is composite, then if the "numerator" is a cubic residue mod the "denominator" the symbol will equal 1, if the symbol does not equal 1 then the "numerator" is a cubic non-residue, but the symbol can equal 1 when the "numerator" is a non-residue:
- Euler, Tractatus ..., §§ 407–410
- Gauss, DA, footnote to art. 358
- Gauss, Theorematis fundamentalis ...
- Gauss, BQ, § 30
- Cox, pp. 83–90
- Lemmermeyer, pp. 199–201, 222–224
- Lemmermeyer, p. 200
- Jacobi, De residuis cubicis ....
- Eisenstein, Beweis des Reciprocitätssatzes ...
- Eisenstein, Nachtrag zum cubischen...
- Eisenstein, Application de l'algèbre...
- cf. Gauss, BQ § 2
- Gauss, DA, Art. 182
- Cox, Ex. 1.4–1.5
- Ireland & Rosen, Props 8.3.1 & 8.3.2
- Euler, Tractatus, §§ 407–401
- Lemmermeyer, p. 222–223
- Tractatus de numerorum doctrina capita sedecim, quae supersunt, 411, footnote (chapter 11)
- Cox, p. 2, Thm. 4.15, Ex. 4.15
- Ireland & Rosen, Prop. 9.6.2, Ex 9.23
- Lemmermeyer, Prop. 7.1 & 7.2
- Gauss, DA footnote to art. 358
- Lemmermeyer, Ex. 7.9
- Jacobi, De residuis cubicis...
- Lemmermeyer, Prop.7.4
- Lemmermeyer, pp. 209–212, Props 7.1–7.3
- Lemmermeyer, Ex. 7.11
- Lemmermeyer, Ex. 7.12
- Gauss, BQ, § 30, translation in Cox, p. 83
- Gauss, BQ, § 30, translation in Cox, p. 84
- Ireland & Rosen p. 14
- Ireland & Rosen Prop 9.1.4
- cf. Gauss, BQ, §§ 38–45
- cf. Gauss, BQ, §§ 46–47
- Ireland & Rosen. Prop. 9.3.1
- Ireland & Rosen, p. 112
- Ireland & Rosen, Prop. 9.3.3
- Ireland & Rosen, Prop. 9.3.4
- Lemmermeyer, Prop 7.7
- Lemmermeyer, Th. 6.9
- Ireland & Rosen, Ex. 9.32–9.37
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.
- Euler, Leonhard (1849), Tractatus de numeroroum doctrina capita sedecim quae supersunt, Comment. Arithmet. 2
This was actually written 1748–1750, but was only published posthumously; It is in Vol V, pp. 182–283 of
- Euler, Leonhard (1911–1944), Opera Omnia, Series prima, Vols I–V, Leipzig & Berlin: Teubner
The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § n". Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n".
- Gauss, Carl Friedrich (1828), Theoria residuorum biquadraticorum, Commentatio prima, Göttingen: Comment. Soc. regiae sci, Göttingen 6
- Gauss, Carl Friedrich (1832), Theoria residuorum biquadraticorum, Commentatio secunda, Göttingen: Comment. Soc. regiae sci, Göttingen 7
These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148
Gauss's fifth and sixth proofs of quadratic reciprocity are in
- Gauss, Carl Friedrich (1818), Theoramatis fundamentalis in doctrina de residuis quadraticis demonstrationes et amplicationes novae
This is in Gauss's Werke, Vol II, pp. 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.
- Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8
- Eisenstein, Ferdinand Gotthold (1844), Beweis des Reciprocitätssatzes für die cubischen Reste in der Theorie der aus den dritten Wurzeln der Einheit zusammengesetzen Zahlen, J. Reine Angew. Math. 27, pp. 289–310 (Crelle's Journal)
- Eisenstein, Ferdinand Gotthold (1844), Nachtrag zum cubischen Reciprocitätssatzes für die aus den dritten Wurzeln der Einheit zusammengesetzen Zahlen, Criterien des cubischen Characters der Zahl 3 and ihrer Teiler, J. Reine Angew. Math. 28, pp. 28–35 (Crelle's Journal)
- Eisenstein, Ferdinand Gotthold (1845), Application de l'algèbre à l'arithmétique transcendante, J. Reine Angew. Math. 29 pp. 177–184 (Crelle's Journal)
These papers are all in Vol I of his Werke.
- Jacobi, Carl Gustave Jacob (1827), De residuis cubicis commentatio numerosa, J. Reine Angew. Math. 2 pp. 66–69 (Crelle's Journal)
This is in Vol VI of his Werke
- Cox, David A. (1989), Primes of the form x2 + n y2, New York: Wiley, ISBN 0-471-50654-0
- Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X