📂Number Theory

Eisenstein Integer

Definition

Equation $\mathbb{Z} [ \omega ] := \left\{ a + \omega b : a, b \in \mathbb{Z} \right\}$ is called the Eisenstein Ring, and its elements are referred to as Eisenstein Integers.

Theorem

• [1]: $\overline{ \omega } = \omega^{2} = - (1 + \omega)$
• [2]: $( a \pm \omega b ) + ( c \pm \omega d) = (a \pm c) + \omega (b \pm d)$
• [3]: $( a + \omega b )( c + \omega d) = (ac - bd) + \omega (ad - bd + bc)$

Description

$\omega$ represents the complex roots $\displaystyle \omega := {{-1 + \sqrt{-3}} \over {2}} = e^{2 \pi i/3 }$ of the cubic equation $x^3 +1 = 0$, and $\mathbb{Z} [\omega]$ is a simple extension of the integer ring $\mathbb{Z}$. As interesting as Gaussian Integers with a bit more complex calculations, it essentially shares similarities with Gaussian Integers, making it not too unfamiliar. In the complex plane, $i$ forms a square lattice with $1,i,-1,-i$, and $\omega$ forms a triangular lattice with $1, - \omega^2, \omega, -1, \omega^2, -\omega$.

Just like there are Gaussian Primes for Gaussian Integers, there are also Eisenstein Primes for Eisenstein Integers.

On the field of $\mathbb{Z} [i]$, the following conventional formula manipulation is possible: $$\begin{align*} (7 + \omega 2)(4 - \omega 2) =& (28 + 4) + \omega (- 14 +4 +8 ) \\ =& 32 - \omega 2 \end{align*}$$ Also, given a natural number $n \in \mathbb{N}$, one can consider $\mathbb{Z}_{n}[i]$ for a finite ring $\mathbb{Z}_{n}$ as well. For example, when stating $n = 7$, the development changes as follows: $$\begin{align*} (7 + i2)(4 -i 2) =& (28 + 4) + \omega (- 14 +4 +8 ) \\ =& 32 - \omega 2 \\ & \equiv 4 - \omega 2 \pmod{7} \\ & \equiv 4 + \omega 5 \pmod{7} \end{align*}$$ Note how naturally the use of congruence is employed. If it works in $\mathbb{Z} [i]$, it seems only logical that it works in $\mathbb{Z} [\omega]$ as well. Just like repeatedly powering $i$ does not produce higher terms, likewise, $\omega$ can also lower the degree as in $\omega^2 = -(1+\omega)$. Of course, this is not merely computation but a fact mathematically guaranteed by the properties of a simple extension.

Meanwhile, $2$ and $3$ are the smallest even and odd primes respectively. Interestingly, the deeper one delves into the properties of Gaussian Integers, the more one becomes fixated on $i$, and similarly, with Eisenstein Integers on $3$. Interestingly enough, $\overline{i} = i^3$ and it is found that $\overline{\omega} = \omega^2$, showcasing the beauty of pure mathematics the more one looks into it.

The zero-divisor graph of the Eisenstein Ring was studied by Alcam.

Proof

[1]

By the definition of $\omega$ and the properties of conjugate $$\begin{align*} \omega^2 =& \left( e^{2 \pi i/3 } \right)^2 \\ =& - e^{ \pi i/3 } \\ =& - {{ 1 + i \sqrt{3} } \over { 2 }} \\ =& \overline{ \left( { -1 + i \sqrt{3} } \over { 2 } \right) } \\ =& \overline{ \omega } \\ =& - (1 + \omega) \end{align*}$$

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[2]

Since $\mathbb{Z} [ \omega ]$ is a ring, and associative and commutative laws hold for addition $$\begin{align*} ( a \pm \omega b ) + ( c \pm \omega d) =& a \pm \omega b + c \pm \omega d \\ =& a \pm c + \omega b \pm \omega d \\ =& (a \pm c) + \omega (b \pm d) \end{align*}$$

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[3]

According to theorems [1] and [2] $$\begin{align*} ( a + \omega b )( c + \omega d) =& ac + \omega ad + \omega bc -(1 + \omega) bd \\ =& (ac - bd) + \omega (ad - bd + bc) \end{align*}$$

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