= Euclidean norm on the Eisenstein integers
{c}
Every <complex number> can be written uniquely as $x+y\omega$ with real $x,y$, where $\omega=(-1+i\sqrt3)/2$. Round $x,y$ separately to <integers> $m,n$. The error is $s+t\omega$ with $|s|,|t|\leq1/2$, and
$$
|s+t\omega|^2=s^2-st+t^2\leq s^2+|st|+t^2\leq\frac34<1.
$$
For <Eisenstein integers> $\alpha,\beta$ with $\beta\ne0$, apply this to $\alpha/\beta$ to choose $q=m+n\omega$. The remainder $r=\alpha-q\beta$ is an <Eisenstein integer> and $N(r)\leq3N(\beta)/4<N(\beta)$ if nonzero. Thus the <Eisenstein norm> is a <Euclidean function> and these integers form a <Euclidean domain>. Coordinate rounding suffices; no closest-point theorem is needed.
\Image[/past-exam-of-the-rings-and-modules-course-of-the-university-of-oxford/2017-eisenstein-division.png]
{title=Coordinate rounding in the <Eisenstein integer> lattice}
{description=The translated error lies in the blue parallelogram, whose farthest corners have squared modulus $3/4$. The dashed unit circle lies strictly outside the error bound.}
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