Cubic reciprocity is a collection of theorems inelementary andalgebraicnumber theory that state conditions under which thecongruencex³ ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of themain theorem, which states that ifp andq are primary numbers in the ring ofEisenstein integers, both coprime to 3, the congruencex³ ≡p (modq) is solvable if and only ifx³ ≡q (modp) is solvable.

Sometime before 1748Euler made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, 62 years after his death.^[1]

Gauss's published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in theDisquisitiones Arithmeticae (1801).^[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 andGaussian sums, respectively) can be applied to cubic and biquadratic reciprocity. Finally, a footnote in the second (of two) monographs onbiquadratic reciprocity (1832) states that cubic reciprocity is most easily described in the ring of Eisenstein integers.^[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From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of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.^[8] In his Königsberg lectures of 1836–37 Jacobi presented proofs.^[7] The first published proofs were by Eisenstein (1844).^[9][10][11]

Acubic residue (modp) is any number congruent to the third power of an integer (modp). Ifx³ ≡a (modp) does not have an integer solution,a is acubic nonresidue (modp).^[12]

Cubic residues are usually only defined in modulusn such that${\displaystyle \lambda (n)}$ (theCarmichael lambda function ofn) is divisible by 3, since for other integern, all residues are cubic residues.

As is often the case in number theory, it is easier to work modulo prime numbers, so in this section all modulip,q, etc., are assumed to be positive odd primes.^[12]

We first note that ifq ≡ 2 (mod 3) is a prime then every number is a cubic residue moduloq. Letq = 3n + 2; since 0 = 0³ is obviously a cubic residue, assumex is not divisible byq. Then byFermat's little theorem,

${\displaystyle x^q\equiv xmodq,x^{q-1}\equiv 1modq}$

Multiplying the two congruences we have

${\displaystyle x^{2q-1}\equiv xmodq}$

Now substituting 3n + 2 forq we have:

${\displaystyle x^{2q-1}=x^{6n+3}={\left(x^{2n+1}\right)}^3.}$

Therefore, the only interesting case is when the modulusp ≡ 1 (mod 3). In this case the non-zero residue classes (modp) can be divided into three sets, each containing (p−1)/3 numbers. Lete be a cubic non-residue. The first set is the cubic residues; the second one ise times the numbers in the first set, and the third ise² times the numbers in the first set. Another way to describe this division is to lete be aprimitive root (modp); 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 ofgroup theory, the cubic residues form a subgroup ofindex 3 of the multiplicative group${\displaystyle (\mathbb{Z}/p\mathbb{Z}{)}^{\times }}$ and the three sets are its cosets.

A theorem of Fermat^[13][14] states that every primep ≡ 1 (mod 3) can be written asp =a² + 3b² and (except for the signs ofa andb) this representation is unique.

Lettingm =a +b andn =a −b, we see that this is equivalent top =m² −mn +n² (which equals (n −m)² − (n −m)n +n² =m² +m(n −m) + (n −m)², som andn are not determined uniquely). Thus,

and it is a straightforward exercise to show that exactly one ofm,n, orm −n is a multiple of 3, so

${\displaystyle p=\frac{1}{4}(L^2+27M^2),}$

and this representation is unique up to the signs ofL andM.^[15]

For relatively prime integersm andn define therational cubic residue symbol as

It is important to note that this symbol doesnot have the multiplicative properties of the Legendre symbol; for this, we need the true cubic character defined below.

Euler's Conjectures. Letp =a² + 3b² be a prime. Then the following hold:^[16][17][18]

The first two can be restated as follows. Letp be a prime that is congruent to 1 modulo 3. Then:^[19][20][21]

• 2 is a cubic residue ofp if and only ifp =a² + 27b².
• 3 is a cubic residue ofp if and only if 4p =a² + 243b².

Gauss's Theorem. Letp be a positive prime such that

${\displaystyle p=3n+1=\frac{1}{4}\left(L^2+27M^2\right).}$

Then${\displaystyle L(n!{)}^3\equiv 1modp.}$^[22][23]

One can easily see that Gauss's Theorem implies:

${\displaystyle {\left[\frac{L}{p}\right]}_3={\left[\frac{M}{p}\right]}_3=1.}$

Jacobi's Theorem (stated without proof).^[24] Letq ≡p ≡ 1 (mod 6) be positive primes. Obviously bothp andq are also congruent to 1 modulo 3, therefore assume:

${\displaystyle p=\frac{1}{4}\left(L^2+27M^2\right),q=\frac{1}{4}\left(L^{\prime 2}+27M^{\prime 2}\right).}$

Letx be a solution ofx² ≡ −3 (modq). Then

${\displaystyle x\equiv \pm \frac{L^{\prime }}{3M^{\prime }}modq,}$

and we have:

Lehmer's Theorem. Letq andp be primes, with${\displaystyle p=\frac{1}{4}\left(L^2+27M^2\right).}$ Then:^[25]

${\displaystyle {\left[\frac{q}{p}\right]}_3=1⟺q\mid LM\text{ or }L\equiv \pm \frac{9r}{2u+1}Mmodq,}$

where

${\displaystyle u\not\equiv 0,1,-\frac{1}{2},-\frac{1}{3}modq\text{and}3u+1\equiv r^2(3u-3)modq.}$

Note that the first condition implies: that any number that dividesL orM is a cubic residue (modp).

The first few examples^[26] of this are equivalent to Euler's conjectures:

Since obviouslyL ≡M (mod 2), the criterion forq = 2 can be simplified as:

${\displaystyle {\left[\frac{2}{p}\right]}_3=1⟺M\equiv 0mod2.}$

Martinet's theorem. Letp ≡q ≡ 1 (mod 3) be primes,${\displaystyle pq=\frac{1}{4}(L^2+27M^2).}$ Then^[27]

${\displaystyle {\left[\frac{L}{p}\right]}_3{\left[\frac{L}{q}\right]}_3=1⟺{\left[\frac{q}{p}\right]}_3{\left[\frac{p}{q}\right]}_3=1.}$

Sharifi's theorem. Letp = 1 + 3x + 9x² be a prime. Then any divisor ofx is a cubic residue (modp).^[28]

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 toimaginary numbers, so that without restriction, the numbers of the forma +bi constitute the object of study ... we call such numbersintegral complex numbers.^[29] [bold in the original]

These numbers are now called thering ofGaussian integers, denoted byZ[i]. Note thati is a fourth root of 1.

In a footnote he adds

The theory of cubic residues must be based in a similar way on a consideration of numbers of the forma +bh whereh is an imaginary root of the equationh³ = 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 ofEisenstein integers. Eisenstein said that to investigate the properties of this ring one need only consult Gauss's work onZ[i] and modify the proofs. This is not surprising since both rings areunique factorization domains.

The "other imaginary quantities" needed for the "theory of residues of higher powers" are therings of integers of thecyclotomic number fields; the Gaussian and Eisenstein integers are the simplest examples of these.

Let

${\displaystyle \omega =\frac{-1+i\sqrt{3}}{2}=e^{\frac{2\pi i}{3}},{\omega }^3=1.}$

And consider the ring ofEisenstein integers:

${\displaystyle \mathbb{Z}[\omega ]=\left\{a+b\omega :a,b\in \mathbb{Z}\right\}.}$

This is aEuclidean domain with thenorm function given by:

${\displaystyle N(a+b\omega )=a^2-ab+b^2.}$

Note that the norm is always congruent to 0 or 1 (mod 3).

Thegroup of units in${\displaystyle \mathbb{Z}[\omega ]}$ (the elements with a multiplicative inverse or equivalently those with unit norm) is a cyclic group of the sixth roots of unity,

${\displaystyle \left\{\pm 1,\pm \omega ,\pm {\omega }^2\right\}.}$

${\displaystyle \mathbb{Z}[\omega ]}$ is aunique factorization domain. The primes fall into three classes:^[32]

• 3 is a special case:

${\displaystyle 3=-{\omega }^2(1-\omega {)}^2.}$

It is the only prime in${\displaystyle \mathbb{Z}}$ divisible by the square of a prime in${\displaystyle \mathbb{Z}[\omega ]}$. The prime 3 is said toramify in${\displaystyle \mathbb{Z}[\omega ]}$.

• Positive primes in${\displaystyle \mathbb{Z}}$ congruent to 2 (mod 3) are also primes in${\displaystyle \mathbb{Z}[\omega ]}$. These primes are said to remaininert in${\displaystyle \mathbb{Z}[\omega ]}$. Note that if${\displaystyle q}$ is any inert prime then:

${\displaystyle N(q)=q^2\equiv 1mod3.}$

• Positive primes in${\displaystyle \mathbb{Z}}$ congruent to 1 (mod 3) are the product of two conjugate primes in${\displaystyle \mathbb{Z}[\omega ]}$. These primes are said tosplit in${\displaystyle \mathbb{Z}[\omega ]}$. Their factorization is given by:

${\displaystyle p=N(\pi )=N(\overline{\pi })=\pi \overline{\pi }.}$

for example

${\displaystyle 7=(3+\omega )(2-\omega ).}$

A number isprimary if it is coprime to 3 and congruent to an ordinary integer modulo${\displaystyle (1-\omega {)}^2,}$ which is the same as saying it is congruent to${\displaystyle \pm 2}$ modulo 3. If${\displaystyle gcd(N(\lambda ),3)=1}$ one of${\displaystyle \lambda ,\omega \lambda ,}$ or${\displaystyle {\omega }^2\lambda }$ 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${\displaystyle \mathbb{Z}[\omega ]}$ is: if${\displaystyle \lambda \ne 0,}$ then

${\displaystyle \lambda =\pm {\omega }^{\mu }(1-\omega {)}^{\nu }{\pi }_1^{{\alpha }_1}{\pi }_2^{{\alpha }_2}{\pi }_3^{{\alpha }_3}\cdots ,\mu \in \{0,1,2\},\nu ,{\alpha }_1,{\alpha }_2,\dots ⩾0}$

where each${\displaystyle {\pi }_i}$ is a primary (under Eisenstein's definition) prime. And this representation is unique, up to the order of the factors.

The notions ofcongruence^[33] andgreatest common divisor^[34] are defined the same way in${\displaystyle \mathbb{Z}[\omega ]}$ as they are for the ordinary integers${\displaystyle \mathbb{Z}}$. Because the units divide all numbers, a congruence modulo${\displaystyle \lambda }$ is also true modulo any associate of${\displaystyle \lambda }$, and any associate of a GCD is also a GCD.

An analogue ofFermat's little theorem is true in${\displaystyle \mathbb{Z}[\omega ]}$: if${\displaystyle \alpha }$ is not divisible by a prime${\displaystyle \pi }$,^[35]

${\displaystyle {\alpha }^{N(\pi )-1}\equiv 1mod\pi .}$

Now assume that${\displaystyle N(\pi )\ne 3}$ so that${\displaystyle N(\pi )\equiv 1mod3.}$ Or put differently${\displaystyle 3\mid N(\pi )-1.}$ Then we can write:

${\displaystyle {\alpha }^{\frac{N(\pi )-1}{3}}\equiv {\omega }^{k}mod\pi ,}$

for a unique unit${\displaystyle {\omega }^k.}$ This unit is called thecubic residue character of${\displaystyle \alpha }$ modulo${\displaystyle \pi }$ and is denoted by^[36]

${\displaystyle {\left(\frac{\alpha }{\pi }\right)}_3={\omega }^k\equiv {\alpha }^{\frac{N(\pi )-1}{3}}mod\pi .}$

The cubic residue character has formal properties similar to those of theLegendre symbol:

• If${\displaystyle \alpha \equiv \beta mod\pi }$ then${\displaystyle {\left(\frac{\alpha }{\pi }\right)}_3={\left(\frac{\beta }{\pi }\right)}_3.}$
• ${\displaystyle {\left(\frac{\alpha \beta }{\pi }\right)}_3={\left(\frac{\alpha }{\pi }\right)}_3{\left(\frac{\beta }{\pi }\right)}_3.}$
• ${\displaystyle \overline{{\left(\frac{\alpha }{\pi }\right)}_3}={\left(\frac{\overline{\alpha }}{\overline{\pi }}\right)}_3,}$ where the bar denotes complex conjugation.
• If${\displaystyle \pi }$ and${\displaystyle \theta }$ are associates then${\displaystyle {\left(\frac{\alpha }{\pi }\right)}_3={\left(\frac{\alpha }{\theta }\right)}_3}$
• The congruence${\displaystyle x^3\equiv \alpha mod\pi }$ has a solution in${\displaystyle \mathbb{Z}[\omega ]}$ if and only if${\displaystyle {\left(\frac{\alpha }{\pi }\right)}_3=1.}$^[37]
• If${\displaystyle a,b\in \mathbb{Z}}$ are such that${\displaystyle gcd(a,b)=gcd(b,3)=1,}$ then${\displaystyle {\left(\frac{a}{b}\right)}_3=1.}$^[38][39]
• 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 theJacobi symbol. As with the Jacobi symbol, this extension sacrifices the "numerator is a cubic residue mod the denominator" meaning: the symbol is still guaranteed to be 1 when the "numerator" is a cubic residue, but the converse no longer holds.

${\displaystyle {\left(\frac{\alpha }{\lambda }\right)}_3={\left(\frac{\alpha }{{\pi }_1}\right)}_3^{{\alpha }_1}{\left(\frac{\alpha }{{\pi }_2}\right)}_3^{{\alpha }_2}\cdots ,}$

where

${\displaystyle \lambda ={\pi }_1^{{\alpha }_1}{\pi }_2^{{\alpha }_2}{\pi }_3^{{\alpha }_3}\cdots }$

Let α and β be primary. Then

${\displaystyle (\frac{\alpha }{\beta }{)}_3=(\frac{\beta }{\alpha }{)}_3.}$

There are supplementary theorems^[40][41] for the units and the prime 1 − ω:

Let α =a +bω be primary,a = 3m + 1 andb = 3n. (Ifa ≡ 2 (mod 3) replace α with its associate −α; this will not change the value of the cubic characters.) Then

${\displaystyle (\frac{\omega }{\alpha }{)}_3={\omega }^{\frac{1-a-b}{3}}={\omega }^{-m-n},(\frac{1-\omega }{\alpha }{)}_3={\omega }^{\frac{a-1}{3}}={\omega }^m,(\frac{3}{\alpha }{)}_3={\omega }^{\frac{b}{3}}={\omega }^n.}$

• Quadratic reciprocity
• Quartic reciprocity
• Octic reciprocity
• Eisenstein reciprocity
• Artin reciprocity

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 theDisquisitiones 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'sWerke, 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'sWerke, Vol II, pp. 47–64

German translations of all three of the above are the following, which also has theDisquisitiones Arithmeticae and Gauss's other papers on number theory.

• Gauss, Carl Friedrich (1965),Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition), translated by Maser, H., 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 hisWerke.

• 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 hisWerke.

• 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

• Lemmermeyer, Franz (2000),Reciprocity Laws: from Euler to Eisenstein, Berlin:Springer,ISBN 3-540-66957-4

• Weisstein, Eric W."Cubic Reciprocity Theorem".MathWorld.

• Algebraic number theory
• Modular arithmetic
• Theorems in number theory

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