Eisenstein Integer

Complex number whose mapping on a coordinate plane produces a triangular lattice

"Eulerian integer" and "Euler integer" redirect here. For other uses, see List of topics named after Leonhard Euler § Euler's numbers.

In the grand, often tedious, landscape of mathematics, we encounter the Eisenstein integers – a construct named, rather predictably, after Gotthold Eisenstein. Though, on occasion, some sources, like Surányi (1997) and Szalay (1991), have seen fit to dub them "Eulerian integers," acknowledging Leonhard Euler's earlier, perhaps less formalized, engagements with these numbers. It seems credit is a fickle thing, even in the realm of absolute truths.

These particular complex numbers manifest in the somewhat rigid form of:

$z = a + b\omega$

Here, a and b are, as one might expect, ordinary integers. The true character, however, is bestowed by ω (omega), which isn't just any old complex number. No, ω is a very specific, rather opinionated entity:

$\omega = \frac{-1+i\sqrt{3}}{2} = e^{i2\pi/3}$

This ω is a primitive, and crucially, non-real, cube root of unity. It satisfies the equation x³ = 1 but x ≠ 1. Its geometric placement on the complex plane is quite precise, sitting at 120 degrees or 2π/3 radians from the positive real axis. This isn't just a trivial detail; it's the very foundation upon which the Eisenstein integers build their unique structure.

Eisenstein integers as the points of a certain triangular lattice in the complex plane

Unlike their more pedestrian cousins, the Gaussian integers, which are content to form a rather unimaginative square lattice in the complex plane, the Eisenstein integers carve out a more intricate, dare I say, slightly more aesthetically pleasing, triangular lattice. Imagine a tessellation of equilateral triangles, and you're getting close to their arrangement. Each point where the vertices of these triangles meet represents an Eisenstein integer. This distinction in lattice structure isn't merely decorative; it fundamentally alters their algebraic behavior and prime factorization properties. Despite their structured arrangement, the collection of all Eisenstein integers remains a countably infinite set, a vast but ultimately enumerable collection of points stretching across the complex plane, much like every other infinite set that manages to be both endless and predictable.

Properties

The Eisenstein integers aren't just a random assortment of complex numbers; they possess a rather specific internal consistency. They coalesce into a commutative ring of algebraic integers within the algebraic number field Q(ω). This field, often referred to as the third cyclotomic field, is formed by adjoining ω to the field of rational numbers. The fact that they are algebraic integers means that each Eisenstein integer z = a + bω is a root of some monic polynomial with integer coefficients. For instance, any z = a + bω is a root of the surprisingly compact quadratic equation:

$z^2 - (2a-b)z + (a^2 - ab + b^2) = 0$

This polynomial demonstrates their algebraic integer status quite elegantly. More fundamentally, ω itself adheres to an even simpler, yet profoundly significant, polynomial identity:

$\omega^2 + \omega + 1 = 0$

This relation is not just a curious fact; it's the algebraic bedrock for all subsequent manipulations and properties of Eisenstein integers. It's what gives ω its unique character and allows for the neat simplification of products.

Speaking of products, when you multiply two Eisenstein integers, say (a + bω) and (c + dω), the result remains firmly within the set of Eisenstein integers. Thanks to the identity ω² = -ω - 1 (derived directly from ω² + ω + 1 = 0), the product can be explicitly stated as:

$(a+b\omega)(c+d\omega) = (ac-bd) + (bc+ad-bd)\omega$

This formula is a testament to the closure property of the ring, ensuring that these numbers behave predictably under multiplication.

The 2-norm of an Eisenstein integer is a critical concept, essentially providing a measure of its "size" in a way that aligns with its algebraic properties. It's simply the squared modulus of the number, and it always yields a positive ordinary (rational) integer. For a + bω, the norm is calculated as:

${\left|a+b\omega\right|}^{2} = {(a-{\tfrac{1}{2}}b)}^{2}+{\tfrac{3}{4}}b^{2} = a^{2}-ab+b^{2}$

This specific form, a² - ab + b², frequently appears in number theory and is directly linked to the quadratic form associated with the cyclotomic field Q(ω). It's a fundamental tool for studying divisibility and factorization within this ring.

Furthermore, the complex conjugate of ω holds a rather neat relationship with ω itself. Given ω = (-1 + i√3)/2, its conjugate is ω̅ = (-1 - i√3)/2. This is precisely ω², which, again, follows directly from ω² + ω + 1 = 0. So, we have:

$\bar{\omega} = \omega^2$

Finally, the group of units within this ring—those Eisenstein integers whose reciprocals are also Eisenstein integers—is a cyclic group. These are precisely the six roots of unity that reside in the complex plane: {±1, ±ω, ±ω²}. Each of these units has a norm of 1, which makes them essentially the "multiplicative identity" in various forms, allowing for re-scaling without changing the underlying divisibility properties. They are the rotational symmetries of the triangular lattice itself.

Euclidean domain

The ring formed by Eisenstein integers is not just any ring; it's a Euclidean domain. This is a significant classification, as it implies a rich algebraic structure, including the existence of a Euclidean algorithm and, consequently, unique factorization into primes. The "norm" N that defines this Euclidean property is, conveniently, the squared modulus we just discussed:

$N(a+b\,\omega) = a^2-ab+b^2$

The essence of a Euclidean domain lies in its division algorithm. For any dividend α and a non-zero divisor β (both Eisenstein integers), there exists a quotient κ and a remainder ρ such that:

$\alpha = \kappa\beta + \rho \ \ {\text{ with }}\ \ N(\rho) < N(\beta)$

Crucially, α, β, κ, and ρ are all Eisenstein integers. This property, the ability to always find a remainder "smaller" than the divisor, is what allows the Euclidean algorithm to function. The algorithm, in turn, underpins Euclid's lemma and guarantees the unique factorization of Eisenstein integers into Eisenstein primes – a concept that brings a comforting order to what might otherwise be a chaotic algebraic landscape.

One practical approach to this division algorithm involves first performing the division in the broader field of complex numbers, then expressing the quotient in terms of ω. If α/β = a + bi for rational a, b ∈ Q, we can rewrite this in the basis of 1 and ω:

$\frac{\alpha}{\beta}\ =\ {\tfrac{1}{\ |\beta |^{2}}}\alpha \overline{\beta}\ =\ a+bi\ =\ a+{\tfrac{1}{\sqrt{3}}}b+{\tfrac{2}{\sqrt{3}}}b\omega$

From this, the Eisenstein integer quotient κ is obtained by simply rounding the rational coefficients to the nearest integer:

$\kappa = \left\lfloor a+{\tfrac{1}{\sqrt{3}}}b\right\rceil +\left\lfloor {\tfrac{2}{\sqrt{3}}}b\right\rceil \omega \ \ {\text{ and }}\ \ \rho ={\alpha }-\kappa \beta$

Here, the notation $\lfloor x\rceil$ denotes any of the standard rounding-to-integer functions. The elegance, or perhaps the sheer mathematical stubbornness, of this procedure is that it works.

The reason this method consistently satisfies N(ρ) < N(β), a condition that surprisingly fails for many other quadratic integer rings, is rooted in geometry. Consider the ideal Z[ω]β = Zβ + Zωβ, which acts by translations on the complex plane. A fundamental domain for this action is a 60°–120° rhombus with vertices at 0, β, ωβ, and β + ωβ. Any Eisenstein integer α will inevitably fall within one of the translates of this parallelogram. The quotient κ is then chosen as one of the vertices of this particular translate. The remainder ρ represents the distance from α to this chosen vertex. The maximum possible distance, if κ is chosen as the closest corner, is significantly less than |β|. Specifically, the maximum distance is approximately $\frac{\sqrt{3}}{2}|\beta|$ , which ensures that:

$|\rho|\leq {\tfrac{\sqrt{3}}{2}}|\beta|<|\beta|$

This strict inequality for the norm of the remainder is precisely what qualifies the Eisenstein integers as a Euclidean domain, allowing for all the pleasant properties of unique factorization. It's almost as if the universe decided to be cooperative, just this once.

Eisenstein primes

For the unrelated concept of an Eisenstein prime of a modular curve, see Eisenstein ideal.

When delving into the structure of Eisenstein integers, the concept of an Eisenstein prime naturally arises. Just as with ordinary integers, a prime number is one that cannot be factored into smaller, non-unit components. If x and y are Eisenstein integers, we say that x divides y if there exists another Eisenstein integer z such that y = zx. A non-unit Eisenstein integer x is then designated an Eisenstein prime if its only non-unit divisors are of the form ux, where u is any of the six units {±1, ±ω, ±ω²}. These are the direct analogues of Gaussian primes within the ring of Gaussian integers.

There are, broadly speaking, two distinct categories of Eisenstein primes, each with its own peculiar charm:

Rational primes congruent to 2 mod 3: Any ordinary prime number p from the set of rational integers that satisfies the condition p ≡ 2 (mod 3) remains an Eisenstein prime in the ring of Eisenstein integers. These primes do not factor further in this new domain; they retain their primality, resisting decomposition into complex factors. Examples include 2, 5, 11, and so on.
3 and rational primes congruent to 1 mod 3: The prime number 3 is a special case. It does not remain prime in the Eisenstein integers; instead, it factors. Similarly, any rational prime p that is congruent to 1 (mod 3) can be expressed as the norm x² - xy + y² of some Eisenstein integer x + ωy. This means such a prime p can be factored within the Eisenstein integers into two distinct, non-associate Eisenstein primes: (x + ωy) and (x + ω²y). These factors are precisely the Eisenstein integers whose norm is a rational prime.

It's worth noting the specific factorization of 3. While it's a prime in the rational integers, in the Eisenstein integers, it breaks down as 3 = -(1 + 2ω)². This can also be written as 3 = (1 - ω)(1 - ω²). The factors 1 - ω and 1 - ω² are associates, meaning they differ only by a unit factor. Specifically, 1 - ω = (-ω)(1 - ω²). Due to this specific relationship, 1 - ω (or 1 - ω²) is often considered a special type of prime factor, sometimes called the "ramified prime" in the context of cyclotomic fields, as noted by sources like MathWorld and Cox (1997).

Let's look at some concrete examples. The first few Eisenstein primes of the form 3n - 1 (which are congruent to 2 mod 3) are:

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ... (This sequence, if you're keeping track, is A003627 in the OEIS). These are the "stubborn" primes that refuse to factor further.

Conversely, natural primes that are congruent to 0 or 1 modulo 3 are decidedly not Eisenstein primes; they readily admit nontrivial factorizations in Z[ω]. For instance:

3 = -(1 + 2ω)² (as mentioned, a bit of a special case, but still a factorization).
7 = (3 + ω)(2 - ω). Here, 7 is 1 mod 3, and it splits into two non-associate Eisenstein primes.

In a more general sense, if a natural prime p is congruent to 1 modulo 3, and can thus be expressed in the form p = a² - ab + b² (which is the norm of a + bω), then it factors over Z[ω] as:

p = (a + bω)(( a - b) - bω)

This factorization reveals the underlying structure when these primes "split" in the Eisenstein integers.

Beyond the rational primes, there are also non-real Eisenstein primes. Some examples include:

2 + ω, 3 + ω, 4 + ω, 5 + 2ω, 6 + ω, 7 + ω, 7 + 3ω.

These are not just arbitrary numbers; they are the fundamental building blocks in this domain. Up to conjugacy and unit multiples (because multiplying by a unit doesn't change primality, only orientation), the primes listed above, along with the rational primes 2 and 5 (which remain prime), represent all Eisenstein primes with an absolute value not exceeding 7. The image accompanying this section illustrates these small Eisenstein primes in the complex plane, showing how those on the green axes are associates of natural primes of the form 3n + 2, while others have norms that are 3 or square roots of natural primes of the form 3n + 1.

As of October 2023, the quest for large primes continues to captivate some. The largest known real Eisenstein prime is a colossal number, the 12th-largest known prime overall: 10223 × 2^31172165 + 1. This behemoth was unearthed by Péter Szabolcs and the PrimeGrid distributed computing project. One can only imagine the sheer computational effort involved in verifying such a thing.

Eisenstein series

Moving into the realm of more advanced complex analysis, Eisenstein series are a class of functions with deep connections to modular forms. When we consider the sum of the reciprocals of all Eisenstein integers (excluding zero, naturally, as division by zero tends to complicate things) raised to the fourth power, a rather striking result emerges:

$\sum_{z\in \mathbf{E} \setminus \{0\}}{\frac{1}{z^{4}}}=G_{4}\left(e^{\frac{2\pi i}{3}}\right)=0$

This means that $e^{2\pi i/3}$ (which is, of course, our familiar ω) is a root of the j-invariant, a crucial modular function. This isn't just a happy coincidence; it reflects the deep symmetries inherent in the triangular lattice formed by the Eisenstein integers.

More generally, for any even integer k, the Eisenstein series G_k(τ) (where τ is in the upper half-plane) has a specific behavior at the point τ = ω. Specifically, G_k(ω) = 0 if and only if k is not a multiple of 6:

$G_{k}\left(e^{\frac{2\pi i}{3}}\right)=0 \quad \text{if and only if} \quad k \not\equiv 0{\pmod {6}}$

This property highlights the special role of the Eisenstein integers in the theory of modular forms and elliptic curves, particularly those with equianharmonic properties.

However, when k is a multiple of 6, the sum yields a non-zero, often quite complex, value. For instance, the sum of the reciprocals of all Eisenstein integers (again, excluding 0) raised to the sixth power can be expressed in terms of the venerable gamma function:

$\sum_{z\in \mathbf{E} \setminus \{0\}}{\frac{1}{z^{6}}}=G_{6}\left(e^{\frac{2\pi i}{3}}\right)={\frac{\Gamma (1/3)^{18}}{8960\pi ^{6}}}\approx 5.86303$

Here, E denotes the set of Eisenstein integers, and G₆ is the Eisenstein series of weight 6. These specific values and zeros are not arbitrary; they encode the symmetries and structures of the Eisenstein lattice in a way that only advanced complex analysis can truly appreciate.

Quotient of C by the Eisenstein integers

Consider the complex plane C. If we take the quotient of this plane by the lattice generated by all Eisenstein integers, what emerges is a complex torus of real dimension 2. This isn't just any torus; it's one of two such tori that boast maximal symmetry among all complex tori.

Geometrically, this torus can be visualized by taking a regular hexagon and identifying each of its three pairs of opposite edges. Imagine gluing the top edge to the bottom edge, the top-right to the bottom-left, and the top-left to the bottom-right. The resulting surface, after this topological surgery, is a torus. This process is a direct consequence of the triangular lattice structure of the Eisenstein integers, where the fundamental parallelogram is a rhombus formed by 0, 1, ω, and 1 + ω, which can be seen as two equilateral triangles joined together. A regular hexagon is a natural choice for a fundamental domain in this lattice.

The other maximally symmetric torus, for comparison, arises from the quotient of the complex plane by the additive lattice of Gaussian integers. This one is conceptually simpler, obtained by identifying each of the two pairs of opposite sides of a square fundamental domain, such as the unit square [0, 1] × [0, 1]. The contrast between the hexagonal and square fundamental domains neatly encapsulates the difference in symmetry between the Eisenstein integers and the Gaussian integers. It's almost as if the universe has a preference for certain shapes when constructing these fundamental spaces.

Notes

Both Surányi, László (1997). Algebra. TYPOTEX. p. 73. and Szalay, Mihály (1991). Számelmélet. Tankönyvkiadó. p. 75. call these numbers "Euler-egészek", that is, Eulerian integers. The latter claims Euler worked with them in a proof.
Weisstein, Eric W. "Eisenstein integer". MathWorld.
Cox, David A. (1997-05-08). Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication (PDF). Wiley. p. 77. ISBN 0-471-19079-9.
" $X^2+X+1$ is reducible in $\mathbb{F}_p[X]$ iff $p\equiv 1{\pmod {3}}$ " .
"Largest Known Primes". The Prime Pages. Retrieved 2023-02-27.
"What are the zeros of the j-function?".
"Show that $G_4(i)\neq 0$ , and $G_6(\rho)\neq 0$ , $\rho=e^{2\pi i/3}$ " .
"Entry 0fda1b – Fungrim: The Mathematical Functions Grimoire". fungrim.org. Retrieved 2023-06-22.
"18.783 Elliptic Curves Lecture 18" (PDF). MIT Mathematics. p. 8. To define a complex structure of $X(1)$ we can restrict attention to $\mathcal{F}^*$ . There are three points that complicate matters: $i,\rho :=e^{2\pi i/3},\infty$ . Lemma Let $G_{\tau}$ be the stabilizer of $\tau \in \mathcal{F}^*$ in $\Gamma$ . Let $S=\left({\begin{array}{cc}0&-1\\1&0\end{array}}\right)$ and $T=\left({\begin{array}{ll}1&1\\0&1\end{array}}\right)$ . Then $G_{\tau }={\begin{cases}\{\pm I\}\simeq \mathbb {Z} /2\mathbb {Z} &{\text{ if }}\tau \notin \{i,\rho ,\infty \}\\\langle S\rangle \simeq \mathbb {Z} /4\mathbb {Z} &{\text{ if }}\tau =i;\\\langle ST\rangle \simeq \mathbb {Z} /6\mathbb {Z} &{\text{ if }}\tau =\rho \\\langle \pm T\rangle \simeq \mathbb {Z} &{\text{ if }}\tau =\infty \end{cases}}$
Weeks, Jeffrey (2001). The Shape of Space. CRC Press. p. 115. 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.

External links

Eisenstein Integer--from MathWorld
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