For questions about Eisenstein integers. The Eisenstein integers are the complex numbers of the form $a+b\omega ,$ where $\omega=e^{2\pi i/3} $ is a primitive $3$rd root of unity. Sometimes they're also called the Eulerian integers.
They form a subring of $\Bbb Q(\omega).$
Eisenstein integers or, Eisenstein-Jacobi integers or, Eulerian integers are defined to be the set $$~\mathbb Z[\omega] = \{a + b\omega : a, b \in \mathbb Z\}~$$ where $$~\omega = \frac{1}{2}(−1 + i \sqrt 3)=e^{2\pi i/3}~.$$
This set lies inside the set of complex numbers $~\mathbb C~$ and they form a commutative ring in the algebraic number field $~\mathbb Q(\omega)~$.
Note:
$1.~$ Like the complex plane is partitioned symmetrically into four quadrants, the Eisenstein integers is symmetrically and radially partitioned into six sextants. Each sextant is defined as follows.
- First sextant: $~ \left\{\eta \in \mathbb Z[\omega] \mid 0 \le \operatorname{Arg}(\eta) <\frac{\pi}{3}\right\} ~$
- Second sextant: $~ \left\{\eta \in \mathbb Z[\omega] \mid \frac\pi3 \le \operatorname{Arg}(\eta) <\frac{2\pi}{3}\right\} ~$
- Third sextant: $~ \left\{\eta \in \mathbb Z[\omega] \mid \frac{2\pi}3 \le \operatorname{Arg}(\eta) <\pi\right\}~$
- Fourth sextant: $~\left\{\eta \in \mathbb Z[\omega] \mid -\pi < \operatorname{Arg}(\eta) < -\frac{2\pi}3 ~~ \text{or,$~~$}\operatorname{Arg}(\eta) = \pi\right\}~$
- Fifth sextant: $~\left\{\eta \in \mathbb Z[\omega] \mid -\frac{2\pi}3 \le \operatorname{Arg}(\eta) <-\frac{\pi}{3}\right\}~$
- Sixth sextant: $~\left\{\eta \in \mathbb Z[\omega] \mid -\frac\pi3 \le \operatorname{Arg}(\eta) <0\right\}~$
$2.~$ Eisenstein integers form a unique factorization domain.
More information can be found in this Wikipedia article.
