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{{Short description|Complex number whose mapping on a coordinate plane produces a triangular lattice}}
{{redirect2|Eulerian integer|Euler integer||List of topics named after Leonhard Euler#Euler's numbers}}

In [[Mathematics|mathematics]], the '''Eisenstein integers''' (named after [[Biography:Gotthold Eisenstein|Gotthold Eisenstein]]), occasionally also known<ref name="euler-name" /> as '''Eulerian integers''' (after [[Biography:Leonhard Euler|Leonhard Euler]]), are the [[Complex number|complex number]]s of the form
: <math>z = a + b\omega ,</math>
where {{math|''a''}} and {{math|''b''}} are [[Integer|integer]]s and
: <math>\omega = \frac{-1 + i\sqrt 3}{2} = e^{i2\pi/3}</math>
is a [[Root of unity#General definition|primitive]] (hence non-real) cube root of unity.

[[File:Eisenstein integer lattice.png|thumb|191px|Eisenstein integers as the points of a certain triangular lattice in the complex plane]]
The Eisenstein integers form a triangular lattice in the [[Complex plane|complex plane]], in contrast with the Gaussian integers, which form a [[Square lattice|square lattice]] in the complex plane. The Eisenstein integers are a [[Countable set|countably infinite set]].

== Properties ==
The Eisenstein integers form a [[Commutative ring|commutative ring]] of [[Algebraic integer|algebraic integer]]s in the [[Algebraic number field|algebraic number field]] {{math|'''Q'''(''ω'')}} – the third [[Cyclotomic field|cyclotomic field]]. To see that the Eisenstein integers are algebraic integers note that each {{math|1=''z'' = ''a'' + ''bω''}} is a root of the [[Monic polynomial|monic polynomial]]
: <math>z^2 - (2a - b)\;\!z + \left(a^2 - ab + b^2\right)~.</math>
In particular, {{math|''ω''}} satisfies the equation
: <math>\omega^2 + \omega + 1 = 0~.</math>

The product of two Eisenstein integers {{math|''a'' + ''bω''}} and {{mvar|''c'' + ''dω''}} is given explicitly by
: <math>(a + b\;\!\omega) \;\! (c + d\;\!\omega)=(ac - bd) + (bc + ad - bd)\;\!\omega~.</math>

The 2-norm of an Eisenstein integer is just its squared modulus, and is given by
: <math>{\left|a + b\;\!\omega\right|}^2 \,= \, {(a - \tfrac{1}{2} b)}^2 + \tfrac{3}{4} b^2 \, = \, a^2 - ab + b^2~,</math>
which is clearly a positive ordinary (rational) integer.

Also, the complex conjugate of {{math|''ω''}} satisfies
: <math>\bar\omega = \omega^2~.</math>

The group of units in this ring is the [[Cyclic group|cyclic group]] formed by the sixth roots of unity in the complex plane: {{math|{{mset|±1, ±''ω'', ±''ω''<sup>2</sup>}}}}, the Eisenstein integers of norm&nbsp;{{math|1}}.

== Euclidean domain ==
The ring of Eisenstein integers forms a [[Euclidean domain]] whose norm {{math|''N''}} is given by the square modulus, as above:
: <math>N(a+b\,\omega) = a^2 - a b + b^2. </math>

A [[Division algorithm|division algorithm]], applied to any dividend {{math|''α''}} and divisor {{math|''β'' ≠ 0}}, gives a quotient {{math|''κ''}} and a remainder {{math|''ρ''}} smaller than the divisor, satisfying:
: <math>\alpha = \kappa \beta +\rho \ \ \text{  with  }\ \ N(\rho) < N(\beta).</math>

Here, {{math|''α''}}, {{math|''β''}}, {{math|''κ''}}, {{math|''ρ''}} are all Eisenstein integers. This algorithm implies the [[Euclidean algorithm]], which proves [[Euclid's lemma]] and the [[Fundamental theorem of arithmetic|unique factorization]] of Eisenstein integers into Eisenstein primes.

One division algorithm is as follows.  First perform the division in the field of complex numbers, and write the quotient in terms of {{math|''ω''}}:
: <math> \frac{\alpha}{\beta}\ =\ \tfrac{1}{\ |\beta|^2}\alpha\overline{\beta} \ =\ a+bi \ =\  a+\tfrac{1}{\sqrt3}b+\tfrac{2}{\sqrt3}b\omega,</math>
for rational {{math|''a'', ''b'' ∈ '''Q'''}}. Then obtain the Eisenstein integer quotient by rounding the rational coefficients to the nearest integer:
: <math>\kappa = \left\lfloor a+\tfrac{1}{\sqrt3}b\right\rceil + \left\lfloor \tfrac{2}{\sqrt3}b\right\rceil\omega \ \ \text{ and }\ \ \rho = {\alpha} - \kappa\beta.</math>
Here <math>\lfloor x\rceil</math> may denote any of the standard [[Rounding|rounding]]-to-integer functions.

The reason this satisfies {{math|''N''(''ρ'') < ''N''(''β'')}}, while the analogous procedure fails for most other [[Quadratic integer|quadratic integer]] rings, is as follows. A [[Fundamental domain|fundamental domain]] for the ideal {{math|1='''Z'''[''ω'']''β'' = '''Z'''''β'' + '''Z'''''ωβ''}}, acting by translations on the complex plane, is the 60°–120° rhombus with vertices {{math|0}}, {{math|''β''}}, {{math|''ωβ''}}, {{math|''β'' + ''ωβ''}}. Any Eisenstein integer {{math|''α''}} lies inside one of the translates of this [[Parallelogram|parallelogram]], and the quotient {{math|''κ''}} is one of its vertices. The remainder is the square distance from {{math|''α''}} to this vertex, but the maximum possible distance in our algorithm is only <math>\tfrac{\sqrt3}2 |\beta|</math>, so <math>|\rho| \leq \tfrac{\sqrt3}2 |\beta|< |\beta|</math>. (The size of {{math|''ρ''}} could be slightly decreased by taking {{math|''κ''}} to be the closest corner.)

== Eisenstein primes ==
{{for|the unrelated concept of an Eisenstein prime of a modular curve|Eisenstein ideal}}

[[File:EisensteinPrimes-01.svg|right|thumb|upright=1.5|Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form {{math|3''n'' + 2}}. All others have an absolute value equal to 3 or square root of a natural prime of the form {{math|3''n'' + 1}}.]]
[[File:Eisenstein primes.svg|right|thumb|upright=1.5|Eisenstein primes in a larger range]]

If {{math|''x''}} and {{math|''y''}} are Eisenstein integers, we say that {{math|''x''}} divides {{math|''y''}} if there is some Eisenstein integer {{math|''z''}} such that {{math|1=''y'' = ''zx''}}. A non-unit Eisenstein integer {{math|''x''}} is said to be an Eisenstein prime if its only non-unit divisors are of the form {{math|''ux''}}, where {{math|''u''}} is any of the six units. They are the corresponding concept to the [[Gaussian prime]]s in the Gaussian integers.

There are two types of Eisenstein prime.
* an ordinary [[Prime number|prime number]] (or ''rational prime'') which is congruent to {{math|2 mod 3}} is also an Eisenstein prime.
* {{math|3}} and each rational prime congruent to {{math|1 mod 3}} are equal to the norm {{math|''x''<sup>2</sup> − ''xy'' + ''y''<sup>2</sup>}} of an Eisenstein integer {{math|''x'' + ''ωy''}}. Thus, such a prime may be factored as {{math|(''x'' + ''ωy'')(''x'' + ''ω''<sup>2</sup>''y'')}}, and these factors are Eisenstein primes: they are precisely the Eisenstein integers whose norm is a rational prime.

In the second type, factors of {{math|3}}, <math>1-\omega</math> and <math>1-\omega^2</math> are associates: <math>1-\omega=(-\omega)(1-\omega^2)</math>, so it is regarded as a special type in some books.<ref>{{MathWorld|EisensteinPrime}}</ref><ref>{{Cite book |title=Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication |last=Cox |first=David A. |date=1997-05-08 |isbn=0-471-19079-9 |pages=77 |publisher=Wiley |url=https://www.math.utoronto.ca/~ila/Cox-Primes_of_the_form_x2+ny2.pdf}}</ref>

The first few Eisenstein primes of the form {{math|3''n'' − 1}} are:
: [[2 (number)|2]], [[5 (number)|5]], [[11 (number)|11]], [[17 (number)|17]], [[23 (number)|23]], [[29 (number)|29]], [[41 (number)|41]], [[47 (number)|47]], [[53 (number)|53]], [[59 (number)|59]], [[71 (number)|71]], [[83 (number)|83]], [[89 (number)|89]], [[101 (number)|101]], ... {{OEIS|id=A003627}}.

Natural primes that are congruent to {{math|0}} or {{math|1}} modulo {{math|3}} are ''not'' Eisenstein primes:<ref>{{Cite web |url=https://math.stackexchange.com/questions/4242234/x2-x-1-is-reducible-in-mathbbf-p-x-iff-p-equiv-1-mod-3 |title=<math>X^2 +X +1</math> is reducible in <math>\mathbb{F}_p [X]</math> iff <math>p\equiv 1\pmod 3</math>}}</ref> they admit nontrivial factorizations in {{math|'''Z'''[''ω'']}}. For example:
: {{math|3 {{=}} −(1 + 2''ω'')<sup>2</sup>}}
: {{math|7 {{=}} (3 + ''ω'')(2 − ''ω'')}}.

In general, if a natural prime {{math|''p''}} is {{math|1}} modulo {{math|3}} and can therefore be written as {{math|''p'' {{=}} ''a''<sup>2</sup> − ''ab'' + ''b''<sup>2</sup>}}, then it factorizes over {{math|'''Z'''[''ω'']}} as
: {{math|p {{=}} (''a'' + ''bω'')((''a'' − ''b'') − ''bω'')}}.

Some non-real Eisenstein primes are
: {{math|2 + ''ω''}}, {{math|3 + ''ω''}}, {{math|4 + ''ω''}}, {{math|5 + 2''ω''}}, {{math|6 + ''ω''}}, {{math|7 + ''ω''}}, {{math|7 + 3''ω''}}.

Up to conjugacy and unit multiples, the primes listed above, together with {{math|2}} and {{math|5}}, are all the Eisenstein primes of [[Absolute value|absolute value]] not exceeding {{math|7}}.

{{As of|2023|10}}, the largest known real Eisenstein prime is the [[Largest known prime number|12th-largest known prime]] {{math|10223 × 2<sup>31172165</sup> + 1}}, discovered by Péter Szabolcs and [[PrimeGrid]].<ref>{{Cite web |title=Largest Known Primes |url=https://primes.utm.edu/top20/page.php?id=3 |access-date=2023-02-27 |website=The [[Prime Pages]]}}</ref>

== Eisenstein series ==
The sum of the reciprocals of all Eisenstein integers excluding {{math|0}} raised to the fourth power is {{math|0}}:<ref>{{Cite web |url=https://math.stackexchange.com/questions/1231810/what-are-the-zeros-of-the-j-function |title=What are the zeros of the j-function?}}</ref>
<math display="block">\sum_{z\in\mathbf{E}\setminus\{0\}}\frac{1}{z^4}=G_4\left(e^{\frac{2\pi i}{3}}\right)=0</math>
so <math>e^{2\pi i/3}</math> is a root of [[J-invariant|j-invariant]].
In general <math>G_k\left(e^{\frac{2\pi i}{3}}\right)=0</math> if and only if <math>k\not\equiv 0 \pmod 6</math>.<ref>{{Cite web |url=https://math.stackexchange.com/questions/2044686/show-that-g-4i-neq-0-and-g-6-rho-neq-0-rho-e2-pi-i-3 |title=Show that <math>G_4(i)\neq 0</math>, and <math>G_6(\rho)\neq 0</math>, <math>\rho=e^{2\pi i /3}</math>}}</ref>

The sum of the reciprocals of all Eisenstein integers excluding {{math|0}} raised to the sixth power can be expressed in terms of the [[Gamma function|gamma function]]:
<math display="block">\sum_{z\in\mathbf{E}\setminus\{0\}}\frac{1}{z^6}=G_6\left(e^{\frac{2\pi i}{3}}\right)=\frac{\Gamma (1/3)^{18}}{8960\pi^6} \approx 5.86303</math>
where {{math|'''E'''}} are the Eisenstein integers and {{math|''G''<sub>6</sub>}} is the [[Eisenstein series]] of weight 6.<ref>{{Cite web |title=Entry 0fda1b – Fungrim: The Mathematical Functions Grimoire |url=https://fungrim.org/entry/0fda1b/ |access-date=2023-06-22 |website=fungrim.org}}</ref>

== Quotient of {{math|'''C'''}} by the Eisenstein integers ==
The [[Quotient ring|quotient]] of the complex plane {{math|'''C'''}} by the [[Lattice (group)|lattice]] containing all Eisenstein integers is a [[Complex torus|complex torus]] of real dimension&nbsp;{{math|2}}. This is one of two tori with maximal [[Symmetry|symmetry]] among all such complex tori.<ref>{{Cite web |title=18.783 Elliptic Curves Lecture 18 |url=https://math.mit.edu/classes/18.783/2022/LectureSlides18.pdf |website=MIT Mathematics |page=8 |quote=To define a complex structure of <math>X(1)</math> we can restrict attention to <math>\mathcal{F}^*</math>. There are three points that complicate matters: <math>i, \rho:=e^{2 \pi i / 3}, \infty</math>.  Lemma Let <math>G_\tau</math> be the stabilizer of <math>\tau \in \mathcal{F}^*</math> in <math>\Gamma</math>. Let <math>S=\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right)</math> and <math>T=\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right)</math>. Then  <math> G_\tau= \begin{cases}\{ \pm I\} \simeq \mathbb{Z} / 2 \mathbb{Z} & \text { if } \tau \notin\{i, \rho, \infty\} \\ \langle S\rangle \simeq \mathbb{Z} / 4 \mathbb{Z} & \text { if } \tau=i ; \\ \langle S T\rangle \simeq \mathbb{Z} / 6 \mathbb{Z} & \text { if } \tau=\rho \\ \langle \pm T\rangle \simeq \mathbb{Z} & \text { if } \tau=\infty\end{cases} </math>}}</ref> This torus can be obtained by identifying each of the three pairs of opposite edges of a regular hexagon.
[[File:Wrapped hexagon topology.png|thumb|center|Identifying each of the three pairs of opposite edges of a regular hexagon.<ref>{{Cite book |title=The Shape of Space |last=Weeks |first=Jeffrey |publisher=CRC Press |year=2001 |pages=115 |quote=Figure 7.13 To physically glue together opposite edges of a hexagon, you must deform the hexagon into the shape of a doughnut surface. A hexagon with abstractly glued edges therefore has the same global topology as a torus.}}</ref>]]
The other maximally symmetric torus is the quotient of the complex plane by the additive lattice of Gaussian integers, and can be obtained by identifying each of the two pairs of opposite sides of a square fundamental domain, such as {{math|[0, 1] × [0, 1]}}.

== See also ==
* [[Gaussian integer]]
* [[Cyclotomic field]]
* [[Systolic geometry]]
* [[Hermite constant]]
* [[Cubic reciprocity]]
* [[Loewner's torus inequality]]
* [[Hurwitz quaternion]]
* [[Quadratic integer]]
* [[Dixon elliptic functions]]
* [[Equianharmonic]]

== Notes ==
{{Reflist|refs=
<ref name="euler-name">Both {{cite book |last=Surányi |first=László |title=Algebra |publisher=TYPOTEX |year=1997 |page=73 }} and {{cite book |last=Szalay |first=Mihály |title=Számelmélet |publisher=Tankönyvkiadó |year=1991 |page=75 }} call these numbers "Euler-egészek", that is, Eulerian integers.  The latter claims Euler worked with them in a proof.</ref>
}}

== External links ==
* [http://mathworld.wolfram.com/EisensteinInteger.html Eisenstein Integer--from MathWorld]

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