Eisenstein integer

For a primitive cube root of unity $\omega$, the Eisenstein integers form the triangular latticeTheir multiplicative norm is $N(a+b\omega )=a^2-ab+b^2$.

The Eisenstein-integer norm is $N(z)=z\overline{z}$. It is multiplicative and takes $a+b\omega$ to $a^2-ab+b^2$.

Every complex number is within squared distance at most $3/4$ of an Eisenstein integer: write it as $x+y\omega$ and round $x$ and $y$ to integers. Therefore, for nonzero $\beta$ and arbitrary $\alpha$, a nearest lattice point $q$ to $\alpha /\beta$ givesThe norm makes the Eisenstein integers a Euclidean domain.