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'''Cubic reciprocity''' is a collection of theorems in [[Number theory#Elementary number theory|elementary]] and [[algebraic number theory|algebraic]]  [[number theory]] that state conditions under which the [[Modular arithmetic|congruence]] ''x''<sup>3</sup>&nbsp;&equiv;&nbsp;''p''&nbsp;(mod&nbsp;''q'') is solvable; the word "reciprocity" comes from the form of the [[#Statement of the theorem|main theorem]], which states that if ''p'' and ''q'' are primary numbers in the ring of [[Eisenstein integer]]s, both coprime to 3,
:'''the congruence ''x''<sup>3</sup> &equiv; ''p'' (mod ''q'') is solvable if and only if  ''x''<sup>3</sup> &equiv; ''q'' (mod ''p'') is.'''

==History==

Sometime before 1748 [[Leonhard Euler|Euler]] made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, after his death.<ref>Euler, ''Tractatus ...'', §§ 407&ndash;410</ref> 

Gauss's published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in the [[Disquisitiones Arithmeticae]] (1801).<ref>Gauss, DA, footnote to  art. 358</ref> In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818)<ref>Gauss, ''Theorematis fundamentalis ...''</ref> he said that he was publishing these proofs because their techniques ([[Gauss's lemma (number theory)|Gauss's lemma]] and [[Quadratic Gauss sum|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.<ref>Gauss, BQ, § 30</ref> 

From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of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.<ref>Cox, pp. 83&ndash;90</ref><ref>Lemmermeyer, pp. 199&ndash;201, 222&ndash;224</ref> Proofs of these were found in his posthumous papers, but it is not clear if they are his or Eisenstein's.<ref name="Lemmermeyer">Lemmermeyer, p. 200</ref>

Jacobi published several theorems about cubic residuacity in 1827, but no proofs.<ref>Jacobi, ''De residuis cubicis ...''.</ref> In his Königsberg lectures of 1836&ndash;37 Jacobi presented proofs.<ref name="Lemmermeyer" /> The first published proofs were by Eisenstein (1844).<ref>Eisenstein, ''Beweis des Reciprocitätssatzes ...''</ref><ref>Eisenstein, ''Nachtrag zum cubischen...''</ref><ref>Eisenstein, ''Application de l'algèbre...''</ref>

==Integers==

A '''cubic residue''' (mod ''p'') is any number congruent to the third power of an integer (mod ''p''). If ''x''<sup>3</sup> &equiv; ''a'' (mod ''p'') does not have an integer solution, ''a'' is a '''cubic nonresidue'''  (mod ''p'').<ref name="CfGauss">cf. Gauss, BQ § 2</ref>

As is often the case in number theory, it is easiest to work modulo prime numbers, so in this section all moduli ''p'', ''q'', etc., are assumed to be positive, odd primes.<ref name="CfGauss" />

The first thing to notice when working within the ring '''Z''' of integers is that if the prime number ''q'' is &equiv; 2 (mod 3) every number is a cubic residue (mod ''q''). Let ''q'' = 3''n'' + 2; since 0 = 0<sup>3</sup> is obviously a cubic residue, assume ''x'' is not divisible by ''q''. Then by [[Fermat's little theorem]], 

:<math>x^q = x^{3 n + 2} \equiv x \pmod{q}\;\mbox{  and  }\;x^{q - 1} = x^{3 n + 1}\equiv  1 \pmod{ q},\mbox{ so }
</math> 
 

:<math>
x = 1 \cdot x \equiv x^q x^{q - 1} = x^{3n + 2} x^{3n+1} = x^{6n + 3} = (x^{2n+1})^3 \pmod{ q}
</math>

is a cubic residue (mod ''q'').  

Therefore, the only interesting case is when the modulus ''p'' &equiv; 1 (mod 3).

In this case, ''p'' &equiv; 1 (mod 3), the nonzero residue classes (mod ''p'') can be divided into three sets, each containing (''p''&minus;1)/3 numbers. Let ''e'' be a cubic nonresidue. The first set is the cubic residues; the second one is ''e'' times the numbers in the first set, and the third is ''e''<sup>2</sup> times the numbers in the first set. Another way to describe this division is to let ''e'' be a [[Primitive root modulo n|primitive root]] (mod ''p''); then the first (respectively second, third) set is the numbers whose indices with respect to this root are &equiv; 0 (resp. 1, 2) (mod 3). In the vocabulary of [[group theory]], the first set is a subgroup of [[Index of a subgroup|index]] 3 (of the multiplicative group &nbsp; '''Z'''/''p'''''Z''' <sup>×</sup>), and the other two are its cosets.

===Primes &equiv; 1 (mod 3)===

A theorem of Fermat<ref>Gauss, DA, Art. 182</ref><ref>Cox, Ex. 1.4&ndash;1.5</ref> states that every prime ''p'' &equiv; 1 (mod 3) is the sum of a square and three times a square: '''''p'' = ''a''<sup>2</sup> + 3''b''<sup>2</sup>''' and (except for the signs of ''a'' and ''b'') this representation is unique. 

Letting ''m'' = ''a'' + ''b'' and  ''n'' = ''a'' &minus; ''b'', we see that this is equivalent to '''''p'' = ''m''<sup>2</sup> &minus; ''mn'' +  ''n''<sup>2</sup>''' (which equals (''n'' &minus; ''m'')<sup>2</sup> &minus; (''n'' &minus; ''m'')''n'' + ''n''<sup>2</sup> = ''m''<sup>2</sup> + ''m''(''n'' &minus; ''m'') + (''n'' &minus; ''m'')<sup>2</sup>, so ''m'' and ''n'' are not determined uniquely). Thus,
:<math>
\begin{align}4p
&= (2m-n)^2 + 3n^2 \\
&= (2n-m)^2 + 3m^2 \\
&= (m+n)^2 + 3(m-n)^2,
\end{align}
</math>
and it is a straightforward exercise to show that exactly one of ''m'', ''n'', or ''m'' &minus; ''n'' is a multiple of 3, so 
:<math>p = \tfrac14 \left(L^2+ 27M^2\right),</math> &nbsp; and this representation is unique up to the signs of ''L'' and ''M''.<ref>Ireland & Rosen, Props 8.3.1 & 8.3.2 </ref>

For relatively prime integers ''m'' and ''n'' define the '''rational cubic residue symbol''' as
:<math>

\left[\frac{m}{n}\right]_3 =
\begin{cases}
&+1 \mbox{ if }m\mbox{ is a cubic residue }\pmod{n}\\
&-1\mbox{ if }m\mbox{ is a cubic nonresidue }\pmod{n}
\end{cases}
</math>

===Euler===

Euler's conjectures<ref>Euler, ''Tractatus'', §§ 407&ndash;401</ref><ref>Lemmermeyer, p. 222&ndash;223<!--; an apparent misprint has been corrected (See Talk)--></ref> are based on the representation ''p'' = ''a''<sup>2</sup> + 3''b''<sup>2</sup>.
:<math>
\begin{align}
\left[\frac{2}{p}\right]_3 =1 &\mbox{ if and only if } 3|b\\
\left[\frac{3}{p}\right]_3 =1 &\mbox{ if and only if } 9|b; \mbox{ or }9|(a\pm b)\\
\left[\frac{5}{p}\right]_3 =1 &\mbox{ if and only if } 15|b; \mbox{ or }3|b \mbox{ and }5|a; \mbox{ or } 15|(a\pm b); \mbox{ or } 15|(2a\pm b)\\
\left[\frac{6}{p}\right]_3 =1 &\mbox{ if and only if } 9|b; \mbox{ or }9|(a\pm 2b)\\
\end{align}
</math><ref>The symbol ''m''|''n'' is read "''m'' divides ''n''" and means there is an ''k'' such that ''n'' = ''m''·''k''.</ref>

The first two can be restated as<ref>Cox, p. 2, Thm. 4.15, Ex. 4.15</ref><ref>Ireland & Rosen, Prop. 9.6.2, Ex 9.23</ref><ref>Lemmermeyer, Prop. 7.1 & 7.2</ref>
* Let ''p'' &equiv; 1 (mod 3) be a positive prime. Then 2 is a cubic residue of  ''p'' if and only if &nbsp;&nbsp;&nbsp;''p'' = ''a''<sup>2</sup> + &nbsp;27''b''<sup>2</sup>.
* Let ''p'' &equiv; 1 (mod 3) be a positive prime. Then 3 is a cubic residue of  ''p'' if and only if &nbsp;4''p'' = ''a''<sup>2</sup> + 243''b''<sup>2</sup>.

<!-- See talk about q=7 -->
Another conjecture of Euler is:<ref>''Tractatus de numerorum doctrina capita sedecim, quae supersunt'', '''411''', footnote (chapter 11) [http://eulerarchive.maa.org/pages/E792.html]</ref>
:<math>
\mbox{ If } \left[\frac{7}{p}\right]_3 =1, \mbox{ then } 
</math>
::<math>
(3|b\mbox{ and }7|a), \mbox{ or } 21|(b\pm a), \mbox{ or } 7|(4b\pm a), \mbox{ or } 21|b, \mbox{ or } 7|(b\pm 2a).
</math>

===Gauss===

Gauss<ref>Gauss, DA footnote to art. 358</ref><ref>Lemmermeyer, Ex. 7.9</ref> proves that if &nbsp;&nbsp;<math>p = 3n + 1= \tfrac14 \left(L^2+ 27M^2\right),</math>&nbsp; then &nbsp;&nbsp;<math> L(n!)^3\equiv 1 \pmod{p},</math> &nbsp;&nbsp; from which &nbsp;<math>\left[\frac{L}{p}\right]_3 = \left[\frac{M}{p}\right]_3 =1</math> is an easy deduction.

===Jacobi===

Jacobi stated (without proof)<ref>Jacobi, ''De residuis cubicis...''</ref>

Let ''q'' &equiv; ''p'' &equiv; 1 (mod 6) be positive primes, &nbsp;&nbsp; <math>p = \tfrac14 \left(L^2+ 27M^2\right),</math> &nbsp;&nbsp; and let ''x'' be a solution of ''x''<sup>2</sup> &equiv; &minus;3 (mod ''q''). Then

:<math>
\left[\frac{q}{p}\right]_3 =1 \mbox{ if and only if } \left[\frac{\frac{L+3Mx}{2}p}{q}\right]_3 =1 \mbox{ if and only if } \left[\frac{(\frac{L+3Mx}{L-3Mx})}{q}\right]_3 =1. 

</math>

(The "numerator" in the last expression is an integer (mod ''q''), not a Legendre symbol).

If &nbsp;&nbsp; <math>q = \tfrac14 \left(L'^2+ 27M'^2\right),</math> &nbsp;&nbsp; then <math>x\equiv\pm \frac{L'}{3M'}\pmod{q}</math>, and we have

:<math>
\left[\frac{q}{p}\right]_3 =1 \mbox{ if and only if } \left[\frac{(\frac{LM'+L'M}{LM'-L'M})}{q}\right]_3 =1. 

</math>

Along the same lines, von Lienen proved<ref>Lemmermeyer, p. 226&ndash;227</ref>

:<math>
\left(\frac{p}{q}\right)_3 \left(\frac{q}{p}\right)_3  = \left(\frac{(\frac{L'M+LM'}{2M})}{p}\right)_3^2.
</math>

In this expression, the definition of <math>\left(\frac{p}{q}\right)_3</math> is different from the one used elsewhere.  To explain this, we split {{math|''p''}} and {{math|''q''}} into <math>p = \pi\overline{\pi}</math>, <math>q = \rho\overline{\rho}</math>, where <math>\pi, \overline{\pi}, \rho, \overline{\rho}</math> are [[Eisenstein prime]]s {{math|≡ 2 (mod 3)}} that satisfy {{math|ρ ≡ ''N''/2''M'' (mod π)}}, {{math|π ≡ ''N''/2''M''&prime; (mod ρ)}}, {{math|''N'' {{=}} ''L''&prime;''M'' + ''LM''&prime;}}, {{math|''L'' ≡ ''L''&prime; ≡ 1 (mod 3)}}.  Von Lienen defined <math>\left(\frac{p}{q}\right)_3</math> as {{math|χ<sub>ρ</sub>(p)}}, where {{math|χ<sub>ρ</sub>(&middot;)}} denotes a cubic residue character modulo {{math|ρ}} (see [[#Cubic residue character|below]]).  Note that the value of {{math|Li(''p'', ''q'') {{=}}}} <math>\left(\frac{p}{q}\right)_3</math> is not uniquely determined.  For example, when <math>p = \alpha\overline{\alpha}</math> satisfies the above conditions, one can choose either <math>\alpha</math> or <math>\overline{\alpha}</math> and call it {{math|π}}.  However, once {{math|π}} is chosen, {{math|ρ ≡ ''N''/2''M'' (mod π)}} is uniquely determined, and both Li(''x'', ''q'') and Li(''x'', ''p'') are unambiguously defined for any ''x''.

Example: {{math|''p'' {{=}} 13 {{=}} (−1+3ω)(−4−3ω)}}, {{math|''q'' {{=}} 79 {{=}} (−7+3ω)(−10−3ω)}}, {{math|''L'' {{=}} −5}}, {{math|''M'' {{=}} 1}}, {{math|''L''&prime; {{=}} −17}}, {{math|''M''&prime; {{=}} 1}}.
* One can choose {{math|π {{=}} −4−3ω}} and {{math|ρ {{=}} −7+3ω}}.  Then {{math|χ<sub>ρ</sub>(''p'') {{=}} ω<sup>2</sup>}}, {{math|χ<sub>π</sub>(''q'') {{=}} 1}}, {{math|χ<sub>π</sub>(''N''/2''M'') {{=}} ω}}, satisfying {{math|χ<sub>ρ</sub>(p)&thinsp;χ<sub>π</sub>(''q'') {{=}} (χ<sub>π</sub>(''N''/2''M''))<sup>2</sup>}}, that is {{math|Li(''p'', ''q'')&thinsp;Li(''q'', ''p'') {{=}} (Li(''N''/2''M'', ''p''))<sup>2</sup>}}.
* Alternatively, one can choose {{math|π {{=}} −1+3ω}} and {{math|ρ {{=}} −10−3ω}}.  Then {{math|χ<sub>ρ</sub>(''p'') {{=}} ω, χ<sub>π</sub>(''q'') {{=}} 1, χ<sub>π</sub>(''N''/2''M'') {{=}} ω<sup>2</sup>}}.  These values are different from the previous ones, but they satisfy the same relationship.

===Other theorems===

[[Emma Lehmer]] proved<ref>Lemmermeyer, Prop.7.4</ref>

Let &nbsp;  <math> q\mbox{ and }p = \tfrac14 \left(L^2+ 27M^2\right)</math> &nbsp; be primes. 
<math>\left[\frac{q}{p}\right]_3 = 1
\mbox{ if and only if }
\begin{cases}
q|LM\mbox{ or }\\
L\equiv\pm \frac{9r}{2u+1} M\pmod{q},  \;\;\;\mbox{ where }\\ \;\;\;\;\; u\not\equiv 0,1,-\frac12, -\frac13 \pmod{q}  \;\;\;\mbox{ and } \\
\;\;\;\;\;3u+1 \equiv r^2 (3u-3)\pmod{q}
\end{cases}
</math>

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

The first few examples<ref>Lemmermeyer, pp. 209&ndash;212, Props 7.1-7.3</ref> of this are equivalent to Euler's conjectures:

:<math>
\begin{align}
\left[\frac{2}{p}\right]_3 =1 &\mbox{ if and only if } &L \equiv M &\equiv 0 \pmod{2} \\
\left[\frac{3}{p}\right]_3 =1 &\mbox{ if and only if } &M &\equiv 0 \pmod{3}\\
\left[\frac{5}{p}\right]_3 =1 &\mbox{ if and only if } &LM &\equiv 0 \pmod{5}\\
\left[\frac{7}{p}\right]_3 =1 &\mbox{ if and only if } &LM &\equiv 0 \pmod{7}\\
\end{align}
</math>
<!--
Lemmermeyer says:
\left[\frac{11}{p}\right]_3 =1 &\mbox{ if and only if } &LM(L-3M)(L+3M) &\equiv 0 \pmod{11}\\
\left[\frac{13}{p}\right]_3 =1 &\mbox{ if and only if } &LM(L-2M)(L+2M) &\equiv 0 \pmod{13}
The correct versions seem:
\left[\frac{11}{p}\right]_3 =1 &\mbox{ if and only if } &LM(L-4M)(L+4M) &\equiv 0 \pmod{11}\\
\left[\frac{13}{p}\right]_3 =1 &\mbox{ if and only if } &LM(L-M)(L+M) &\equiv 0 \pmod{13}
See talk.
-->
Since obviously ''L'' ≡ ''M'' (mod 2), the criterion for ''q'' = 2 can be simplified as:
:<math>
\left[\frac{2}{p}\right]_3 =1 \mbox{ if and only if } \qquad\quad M \equiv 0 \pmod{2}
</math>
Martinet proved<ref>Lemmermeyer, Ex. 7.11</ref>

Let ''p'' &equiv; ''q'' &equiv; 1 (mod 3) be primes, &nbsp; <math> pq = \tfrac14 \left(L^2+ 27M^2\right).</math> &nbsp; Then

:<math>
\left[\frac{L}{p}\right]_3  \left[\frac{L}{q}\right]_3 =1 \;\;\mbox{ if and only if } \;\;\left[\frac{q}{p}\right]_3  \left[\frac{p}{q}\right]_3 =1 
</math>

Sharifi proved<ref>Lemmermeyer, Ex. 7.12</ref>

Let '''''p''  = 1 + 3''x'' + 9''x''<sup>2</sup>''' be prime. Then 
:'''Any divisor of ''x'' is a cubic residue (mod ''p'').'''

==Eisenstein integers==

===Background===

In his second monograph on biquadratic reciprocity, Gauss says:

<blockquote>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'''.<ref>Gauss, BQ, § 30, translation in Cox, p. 83</ref> [bold in the original]</blockquote>

These numbers are now called the [[ring (mathematics)|ring]] of [[Gaussian integers]], denoted by '''Z'''[''i'']. Note that ''i'' is a fourth root of 1.

In a footnote he adds

<blockquote>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 ''h''<sup>3</sup> = 1 ... and similarly the theory of residues of higher powers leads to the introduction of other imaginary quantities.<ref>Gauss, BQ, § 30, translation in Cox, p. 84</ref></blockquote>

In his first monograph on cubic reciprocity<ref>Ireland & Rosen p. 14</ref> 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 domain]]s.

The "other imaginary quantities" needed for the "theory of residues of higher powers" are the [[Ring of integers|rings of integers]] of the [[Cyclotomic field|cyclotomic number field]]s; the Gaussian and Eisenstein integers are the simplest examples of these.

===Facts and terminology===

Let <math>\omega = \frac{-1 + i\sqrt 3}{2} = e^\frac{2\pi i}{3}</math> be a complex [[cube root of unity]].  The Eisenstein integers '''Z'''[&omega;] are all numbers of the form ''a'' + ''b''&omega; where ''a'' and ''b'' are ordinary [[integer]]s.

Since &omega;<sup>3</sup> &minus; 1 = (&omega; &minus; 1)(&omega;<sup>2</sup> + &omega; + 1) = 0 and &omega; ≠  1, we have &omega;<sup>2</sup> = &minus; &omega; &minus; 1 and &omega; =  &minus; &omega;<sup>2</sup> &minus; 1. Since  <math>\omega^3 = \omega \omega^2 = \omega \overline{\omega} =1, \;\;\ \overline{\omega} = \omega^2</math> &nbsp; and &nbsp; <math>\overline{\omega^2} = \omega</math> &nbsp; where the bar denotes [[complex conjugation]]. Also, &nbsp; <math>\omega -\overline{\omega} = i\sqrt{3}.</math>

If &lambda; = ''a'' + ''b''&omega; and &mu; = ''c'' + ''d''&omega;,
:&lambda; + &mu; = (''a'' + ''c'') + (''b'' + ''d'')&omega; and
:&lambda; &mu; = ''ac'' + (''ad'' + ''bc'')&omega; + ''bd''&omega;<sup>2</sup> = (''ac'' &minus; ''bd'') + (''ad'' + ''bc'' &minus; ''bd'')&omega;. 

This shows that '''Z'''[&omega;] is closed under addition and multiplication, making it a [[ring (mathematics)|ring]].

The '''units''' are the numbers that divide 1. They are ±1, ±&omega;, and ±&omega;<sup>2</sup>. They are similar to 1 and &minus;1 in the ordinary integers, in that they divide every number. The units are the powers of &minus;&omega;, a '''sixth''' (not just a third) root of unity.

Given a number &lambda; = ''a'' + ''b''&omega;, its '''conjugate''' means  its complex conjugate ''a'' + ''b&omega;<sup>2</sup>'' = (''a'' &minus; ''b'') &minus; ''b''&omega;  &nbsp;(not ''a'' &minus; ''b''&omega;), and its '''associates''' are its six unit multiples:<ref>Ireland & Rosen Prop. 9.3.5</ref>

:<math>
\begin{align}
\lambda &= a + b\omega \\
\omega\lambda &= -b + (a -b)\omega \\
\omega^2\lambda &= (b-a)-a\omega \\
-\lambda &= -a-b\omega \\
-\omega\lambda &= b + (b -a)\omega \\
-\omega^2\lambda &= (a -b) + a\omega
\end{align}
</math>

The '''norm''' of &lambda; = ''a'' + ''b''&omega; is the product of &lambda; and its conjugate <math> \mathrm{N} \lambda = \lambda\overline{\lambda}=a^2-ab+b^2.</math> From the definition, if &lambda; and &mu; are two Eisenstein integers, N&lambda;&mu; = N&lambda;  N&mu;; in other words, the norm is a [[completely multiplicative function]]. The norm of zero is zero, the norm of any other number is a positive integer. &epsilon; is a unit if and only if N&epsilon; = 1. Note that the norm is always &equiv; 0 or &equiv; 1 (mod 3).

'''Z'''[&omega;] is a [[unique factorization domain]]. The primes fall into three classes:<ref>Ireland & Rosen Prop 9.1.4</ref>
* 3 is a special case: 3 = &minus;&omega;<sup>2</sup>(1 &minus; &omega;)<sup>2</sup>. It is the only prime in '''Z''' divisible by the square of a prime in '''Z'''[&omega;]. In algebraic number theory, 3 is said to ramify in '''Z'''[&omega;].

* Positive primes in '''Z''' &equiv; 2 (mod 3) are also primes in '''Z'''[&omega;]. In algebraic number theory, these primes are said to remain inert in '''Z'''[&omega;].

* Positive primes in '''Z''' &equiv; 1 (mod 3) are the product of two conjugate primes in '''Z'''[&omega;]. &nbsp; <math>p=\mathrm{N} \pi = \mathrm{N} \overline{\pi}= \pi  \overline{\pi}</math> &nbsp; In algebraic number theory, these primes are said to split in '''Z'''[&omega;]. 

Thus, inert primes are 2, 5, 11, 17, ... and a factorization of the split primes is 
:&nbsp;7 = (3 + &omega;) × (2 &minus; &omega;), 
:13 = (4 + &omega;) × (3 &minus; &omega;), 
:19 = (3 &minus; 2&omega;) × (5 + 2&omega;), 
:31 = (1 + 6&omega;) × (&minus;5 &minus; 6&omega;), ...

The associates and conjugate of a prime are also primes.

Note that the norm of an inert prime ''q'' is N''q'' = ''q''<sup>2</sup> &equiv; 1 (mod 3).

In order to state the unique factorization theorem, it is necessary to have a way of distinguishing one of the associates of a number. Eisenstein  defines<ref>Ireland & Rosen, p. 113</ref> a number to be '''primary''' if it is &equiv; 2 (mod 3). It is straightforward to show that if gcd(N&lambda;, 3) = 1 then exactly one associate of &lambda; is primary. A disadvantage of this definition is that the product of two primary numbers is the negative of a primary.

Most modern authors<ref name="Ireland">Ireland & Rosen, p. 206</ref><ref>Lemmermeyer, p. 361 calls such numbers ''semi-primary''.</ref> say that a number is primary if it is coprime to 3 and congruent to an ordinary integer (mod (1 &minus; &omega;)<sup>2</sup>), which is the same as saying it is &equiv; ±2 (mod 3). There are two reasons to do  this: first, the product of two primaries is a primary, and second, it generalizes to all cyclotomic number fields.<ref name="Ireland" /> Under this definition, if gcd(N&lambda;, 3) = 1 one of &lambda;, &omega;&lambda;, or &omega;<sup>2</sup>&lambda; is primary. A primary under Eisenstein's definition is primary under the modern one, and if &lambda; is primary under the modern one, either &lambda; or &minus;&lambda; is primary under Eisenstein's. Since &minus;1 is a cube, this does not affect the statement of cubic reciprocity, but it does affect the unique factorization theorem. This article uses the modern definition, so

The product of two primary numbers is primary and the conjugate of a primary number is also primary.

The unique factorization theorem for '''Z'''[&omega;] is: if &lambda; ≠ 0, then
:<math>\lambda = \pm\omega^\mu(1-\omega)^\nu\pi_1^{\alpha_1}\pi_2^{\alpha_2}\pi_3^{\alpha_3} \dots</math>
where &nbsp; 0 &le; &mu; &le; 2, &nbsp; &nu; &ge; 0, &nbsp; each &pi;<sub>''i''</sub> is a primary (under Eisenstein's definition) prime, and each &alpha;<sub>''i''</sub> &ge; 1, and this representation is unique, up to the order of the factors.

The notions of [[Modular arithmetic|congruence]]<ref>cf. Gauss, BQ, §§ 38&ndash;45</ref> and [[greatest common divisor]]<ref>cf. Gauss, BQ, §§ 46&ndash;47</ref> are defined the same way in '''Z'''[&omega;] as they are for the ordinary integers '''Z'''. Because the units divide all numbers, a congruence (mod &lambda;) is also true modulo any associate of &lambda;, and any associate of a GCD is also a GCD.

===Cubic residue character===

An analogue of [[Fermat's little theorem]] is true in '''Z'''[&omega;]: if &alpha; is not divisible by a prime &pi;, <ref>Ireland & Rosen. Prop 9.3.1</ref>
:<math>\alpha^{\mathrm{N}  \pi - 1} \equiv 1 \pmod{\pi}</math>

Now assume that N&pi; ≠ 3, so that N&pi; &equiv; 1 (mod 3).

Then &nbsp; <math>\alpha^{\frac{\mathrm{N} \pi - 1}{3}}</math> &nbsp; &nbsp; makes sense, and <math>\alpha^{\frac{\mathrm{N} \pi - 1}{3}}\equiv \omega^k \pmod{\pi}</math> &nbsp; &nbsp; for a unique unit &omega;<sup>''k''</sup>.

This unit is called the '''cubic residue character''' of &alpha; (mod &pi;) and is denoted by<ref>Ireland & Rosen, p. 112</ref> 
:<math>\left(\frac{\alpha}{\pi}\right)_3 = \omega^k  \equiv \alpha^{\frac{\mathrm{N} \pi - 1}{3}} \pmod{\pi}.</math> 

It has formal properties similar to those of the [[Legendre symbol]].<ref>Ireland & Rosen, Prop 9.3.3</ref>

:The congruence &nbsp;&nbsp; <math>x^3 \equiv \alpha \pmod{\pi}</math> &nbsp;&nbsp; is solvable in '''Z'''[&omega;] if and only if &nbsp;&nbsp;<math>\left(\frac{\alpha}{\pi}\right)_3 = 1.</math><ref>Ireland & Rosen, Prop. 9.3.3</ref>

:<math>\Bigg(\frac{\alpha\beta}{\pi}\Bigg)_3=\Bigg(\frac{\alpha}{\pi}\Bigg)_3\Bigg(\frac{\beta}{\pi}\Bigg)_3</math>

:<math>\overline{\Bigg(\frac{\alpha}{\pi}\Bigg)_3}=\Bigg(\frac{\overline{\alpha}}{\overline{\pi}}\Bigg)_3</math> &nbsp;&nbsp;&nbsp;&nbsp;where the bar denotes complex conjugation.

:if &pi; and &theta; are associates, &nbsp;&nbsp;<math>\Bigg(\frac{\alpha}{\pi}\Bigg)_3=\Bigg(\frac{\alpha}{\theta}\Bigg)_3</math>

:if &alpha; &equiv; &beta; (mod &pi;), &nbsp;&nbsp;<math>\Bigg(\frac{\alpha}{\pi}\Bigg)_3=\Bigg(\frac{\beta}{\pi}\Bigg)_3</math>

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 nonresidue, but the symbol can equal 1 when the "numerator" is a nonresidue:
:<math>\left(\frac{\alpha}{\lambda}\right)_3 = \left(\frac{\alpha}{\pi_1}\right)_3^{\alpha_1} \left(\frac{\alpha}{\pi_2}\right)_3^{\alpha_2} \dots</math>&nbsp;&nbsp; where &nbsp;&nbsp;<math> 
\lambda = \pi_1^{\alpha_1}\pi_2^{\alpha_2}\pi_3^{\alpha_3} \dots</math>

:If ''a'' and ''b'' are ordinary integers,  gcd(''a'', ''b'') = gcd(''b'', 3) = 1, then<ref>Ireland & Rosen, Prop. 9.3.4</ref><ref>Lemmermeyer, Prop 7.7</ref> &nbsp;&nbsp; <math>\left(\frac{a}{b}\right)_3 = 1.</math>

===Statement of the theorem===

Let &alpha; and &beta; be primary. Then

:<math>\Bigg(\frac{\alpha}{\beta}\Bigg)_3 = \Bigg(\frac{\beta}{\alpha}\Bigg)_3. </math>

There are supplementary theorems<ref>Lemmermeyer, Th. 6.9</ref><ref>Ireland & Rosen, Ex. 9.32&ndash;9.37</ref> for the units and the prime 1 &minus; &omega;:

Let &alpha; = ''a'' + ''b''&omega; be primary, ''a'' = 3''m'' + 1 and ''b'' = 3''n''. (If ''a'' &equiv; 2 (mod 3) replace &alpha; with its associate &minus;&alpha;; this will not change the value of the cubic characters.) Then

:<math>
\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.
</math>

==See also==

*[[Quadratic reciprocity]]
*[[Quartic reciprocity]]
*[[Eisenstein reciprocity]]
*[[Artin reciprocity]]

==Notes==
{{reflist|2}}

==References==

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===

*{{citation
  | last1 = Euler | first1 = Leonhard
  | title = Tractatus de numeroroum doctrina capita sedecim quae supersunt
  | publisher = Comment. Arithmet. 2
  | date = 1849}}

This was actually written 1748&ndash;1750, but was only published posthumously; It is in Vol V, pp. 182&ndash;283 of

*{{citation
  | last1 = Euler | first1 = Leonhard
  | title = Opera Omnia, Series prima, Vols I&ndash;V
  | publisher = Teubner
  | location = Leipzig & Berlin
  | date = 1911&ndash;1944}}

===Gauss===

The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1&ndash;23 and the second §§ 24&ndash;76. Footnotes referencing these are of the form "Gauss, BQ, § ''n''". Footnotes referencing the ''Disquisitiones Arithmeticae'' are of the form "Gauss, DA, Art. ''n''".

*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | title = Theoria residuorum biquadraticorum, Commentatio prima
  | publisher = Comment. Soc. regiae sci, Göttingen 6
  | location = Göttingen
  | date = 1828}}

*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | title = Theoria residuorum biquadraticorum, Commentatio secunda
  | publisher = Comment. Soc. regiae sci, Göttingen 7
  | location = Göttingen
  | date = 1832}}

These are in Gauss's ''Werke'', Vol II, pp. 65&ndash;92 and 93&ndash;148

Gauss's fifth and sixth proofs of quadratic reciprocity are in

*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | title = Theoramatis fundamentalis in doctrina de residuis quadraticis demonstrationes et amplicationes novae 
    | date = 1818}}

This is in Gauss's ''Werke'', Vol II, pp. 47&ndash;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.

*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | last2 = Maser | first2 = H. (translator into German)  
  | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)
  | publisher = Chelsea
  | location = New York
  | date = 1965
  | isbn = 0-8284-0191-8}}

===Eisenstein===

*{{citation
  | last1 = Eisenstein | first1 = Ferdinand Gotthold
  | title = Beweis des Reciprocitätssatzes für die cubischen Reste in der Theorie der aus den dritten Wurzeln der Einheit zusammengesetzen Zahlen
  | publisher = J. Reine Angew. Math. 27, pp. 289&ndash;310 (Crelle's Journal)
  | date = 1844}}

*{{citation
  | last1 = Eisenstein | first1 = Ferdinand Gotthold
  | title = Nachtrag zum cubischen Reciprocitätssatzes für die aus den dritten Wurzeln der Einheit zusammengesetzen Zahlen, Criterien des cubischen Characters der Zahl 3 and ihrer Teiler
  | publisher = J. Reine Angew. Math. 28, pp. 28&ndash;35 (Crelle's Journal)
  | date = 1844}}

*{{citation
  | last1 = Eisenstein | first1 = Ferdinand Gotthold
  | title = Application de l'algèbre à l'arithmétique transcendante
  | publisher = J. Reine Angew. Math. 29 pp. 177&ndash;184 (Crelle's Journal)
  | date = 1845}}

These papers are all in Vol I of his ''Werke''.

===Jacobi===

*{{citation
  | last1 = Jacobi | first1 = Carl Gustave Jacob
  | title = De residuis cubicis commentatio numerosa
  | publisher = J. Reine Angew. Math. 2 pp. 66&ndash;69 (Crelle's Journal)
  | date = 1827}}

This is in Vol VI of his ''Werke''

===Modern authors===

*{{citation
  | last1 = Cox  | first1 = David A.
  | title = Primes of the form x<sup>2</sup> + n y<sup>2</sup>
  | publisher = Wiley
  | location = New York
  | date = 1989
  | isbn = 0-471-50654-0}}

*{{citation
  | last1 = Ireland  | first1 = Kenneth
  | last2 = Rosen  | first2 = Michael
  | title = A Classical Introduction to Modern Number Theory (Second edition)
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = New York
  | date = 1990
  | isbn = 0-387-97329-X}}

*{{citation
  | last1 = Lemmermeyer  | first1 = Franz
  | title = Reciprocity Laws: from Euler to Eisenstein
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = Berlin
  | date = 2000
  | isbn = 3-540-66957-4}}

==External links==

* {{mathworld|urlname=CubicReciprocityTheorem|title=Cubic Reciprocity Theorem}}

[[Category:Algebraic number theory]]
[[Category:Modular arithmetic]]
[[Category:Theorems in number theory]]