Cubic reciprocity

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,

${\displaystyle {\begin{aligned}4p&=(2m-n)^{2}+3n^{2}\\&=(2n-m)^{2}+3m^{2}\\&=(m+n)^{2}+3(m-n)^{2}\end{aligned}}}$

and it is a straightforward exercise to show that exactly one of m, n, or m − 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 of L and M.[15]

For relatively prime integers m and n define the rational cubic residue symbol as

${\displaystyle \left[{\frac {m}{n}}\right]_{3}={\begin{cases}1&m{\text{ is a cubic residue }}{\bmod {n}}\\-1&m{\text{ is a cubic non-residue }}{\bmod {n}}\end{cases}}}$

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:[16][17][18]

${\displaystyle {\begin{aligned}\left[{\tfrac {2}{p}}\right]_{3}=1\quad &\Longleftrightarrow \quad 3\mid b\\\left[{\tfrac {3}{p}}\right]_{3}=1\quad &\Longleftrightarrow \quad 9\mid b{\text{ or }}9\mid (a\pm b)\\\left[{\tfrac {5}{p}}\right]_{3}=1\quad &\Longleftrightarrow \quad 15\mid b{\text{ or }}3\mid b{\text{ and }}5\mid a{\text{ or }}15\mid (a\pm b){\text{ or }}15\mid (2a\pm b)\\\left[{\tfrac {6}{p}}\right]_{3}=1\quad &\Longleftrightarrow \quad 9\mid b{\text{ or }}9\mid (a\pm 2b)\\\left[{\tfrac {7}{p}}\right]_{3}=1\quad &\Longrightarrow \quad (3\mid b{\text{ and }}7\mid a){\text{ or }}21\mid (b\pm a){\text{ or }}7\mid (4b\pm a){\text{ or }}21\mid b{\text{ or }}7\mid (b\pm 2a)\end{aligned}}}$

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

• 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

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

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

One can easily see that Gauss's Theorem implies:

${\displaystyle \left[{\tfrac {L}{p}}\right]_{3}=\left[{\tfrac {M}{p}}\right]_{3}=1.}$

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

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

Let x be a solution of x² ≡ −3 (mod q). Then

${\displaystyle x\equiv \pm {\frac {L'}{3M'}}{\bmod {q}},}$

and we have:

${\displaystyle {\begin{aligned}\left[{\frac {q}{p}}\right]_{3}=1\quad &\Longleftrightarrow \quad \left[{\frac {{\frac {L+3Mx}{2}}p}{q}}\right]_{3}=1\quad \Longleftrightarrow \quad \left[{\frac {\frac {L+3Mx}{L-3Mx}}{q}}\right]_{3}=1\\\left[{\frac {q}{p}}\right]_{3}=1\quad &\Longrightarrow \quad \left[{\frac {\frac {LM'+L'M}{LM'-L'M}}{q}}\right]_{3}=1\end{aligned}}}$

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

${\displaystyle \left[{\frac {q}{p}}\right]_{3}=1\quad \Longleftrightarrow \quad q\mid LM{\text{ or }}L\equiv \pm {\frac {9r}{2u+1}}M{\bmod {q}},}$

where

${\displaystyle u\not \equiv 0,1,-{\tfrac {1}{2}},-{\tfrac {1}{3}}{\bmod {q}}\quad {\text{and}}\quad 3u+1\equiv r^{2}(3u-3){\bmod {q}}.}$

Note that the first condition implies: that any number that divides L or M is a cubic residue (mod p).

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

${\displaystyle {\begin{aligned}\left[{\frac {2}{p}}\right]_{3}=1\quad &\Longleftrightarrow \quad L\equiv M\equiv 0{\bmod {2}}\\\left[{\frac {3}{p}}\right]_{3}=1\quad &\Longleftrightarrow \quad M\equiv 0{\bmod {3}}\\\left[{\frac {5}{p}}\right]_{3}=1\quad &\Longleftrightarrow \quad LM\equiv 0{\bmod {5}}\\\left[{\frac {7}{p}}\right]_{3}=1\quad &\Longleftrightarrow \quad LM\equiv 0{\bmod {7}}\end{aligned}}}$

Since obviously L ≡ M (mod 2), the criterion for q = 2 can be simplified as:

${\displaystyle \left[{\frac {2}{p}}\right]_{3}=1\quad \Longleftrightarrow \quad M\equiv 0{\bmod {2}}.}$

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

${\displaystyle \left[{\frac {L}{p}}\right]_{3}\left[{\frac {L}{q}}\right]_{3}=1\quad \Longleftrightarrow \quad \left[{\frac {q}{p}}\right]_{3}\left[{\frac {p}{q}}\right]_{3}=1.}$

Sharifi's theorem. Let p = 1 + 3x + 9x² be a prime. Then any divisor of x is a cubic residue (mod p).[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 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]. Note that i 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 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 (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.

Let

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

And consider the ring of Eisenstein integers:

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

This is a Euclidean domain with the norm 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).

The group 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 a unique 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 to ramify 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 remain inert in ${\displaystyle \mathbb {Z} [\omega ]}$. Note that if ${\displaystyle q}$ is any inert prime then:

${\displaystyle N(q)=q^{2}\equiv 1{\bmod {3}}.}$

• 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 to split 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 is primary 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 ${\displaystyle \lambda \neq 0,}$ then

${\displaystyle \lambda =\pm \omega ^{\mu }(1-\omega )^{\nu }\pi _{1}^{\alpha _{1}}\pi _{2}^{\alpha _{2}}\pi _{3}^{\alpha _{3}}\cdots ,\qquad \mu \in \{0,1,2\},\quad \nu ,\alpha _{1},\alpha _{2},\ldots \geqslant 0}$

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 of congruence[33] and greatest 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 of Fermat's little theorem is true in ${\displaystyle \mathbb {Z} [\omega ]}$: if ${\displaystyle \alpha }$ is not divisible by a prime ${\displaystyle \pi }$,[35]

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

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

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

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

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

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

• If ${\displaystyle \alpha \equiv \beta {\bmod {\pi }}}$ then ${\displaystyle \left({\tfrac {\alpha }{\pi }}\right)_{3}=\left({\tfrac {\beta }{\pi }}\right)_{3}.}$
• ${\displaystyle \left({\tfrac {\alpha \beta }{\pi }}\right)_{3}=\left({\tfrac {\alpha }{\pi }}\right)_{3}\left({\tfrac {\beta }{\pi }}\right)_{3}.}$
• ${\displaystyle {\overline {\left({\tfrac {\alpha }{\pi }}\right)_{3}}}=\left({\tfrac {\overline {\alpha }}{\overline {\pi }}}\right)_{3},}$ where the bar denotes complex conjugation.
• If ${\displaystyle \pi }$ and ${\displaystyle \theta }$ are associates then ${\displaystyle \left({\tfrac {\alpha }{\pi }}\right)_{3}=\left({\tfrac {\alpha }{\theta }}\right)_{3}}$
• The congruence ${\displaystyle x^{3}\equiv \alpha {\bmod {\pi }}}$ has a solution in ${\displaystyle \mathbb {Z} [\omega ]}$ if and only if ${\displaystyle \left({\tfrac {\alpha }{\pi }}\right)_{3}=1.}$[37]
• If ${\displaystyle a,b\in \mathbb {Z} }$ are such that ${\displaystyle \gcd(a,b)=\gcd(b,3)=1,}$ then ${\displaystyle \left({\tfrac {a}{b}}\right)_{3}=1.}$[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 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:

${\displaystyle \left({\frac {\alpha }{\lambda }}\right)_{3}=\left({\frac {\alpha }{\pi _{1}}}\right)_{3}^{\alpha _{1}}\left({\frac {\alpha }{\pi _{2}}}\right)_{3}^{\alpha _{2}}\cdots ,}$

where

${\displaystyle \lambda =\pi _{1}^{\alpha _{1}}\pi _{2}^{\alpha _{2}}\pi _{3}^{\alpha _{3}}\cdots }$

Let α and β be primary. Then

${\displaystyle {\Bigg (}{\frac {\alpha }{\beta }}{\Bigg )}_{3}={\Bigg (}{\frac {\beta }{\alpha }}{\Bigg )}_{3}.}$

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.) Then

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

• 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

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