Euclidean domain

A Euclidean domain admits division with remainder measured by a decreasing Euclidean function.

A Euclidean function on an integral domain $R$ assigns a nonnegative integer to every nonzero element so that, for $a,b\in R$ with $b\ne 0$, there are $q,r\in R$ satisfying $a=bq+r$ and either $r=0$ or $d(r)<d(b)$.

A Gaussian integer has the form $a+bi$ with $a,b\in \mathbb{Z}$. Its multiplicative norm is $N(a+bi)=a^2+b^2$, which makes $\mathbb{Z}[i]$ a Euclidean domain.

If $\alpha =a+bi$ is a Gaussian integer and $P$ is a nonzero prime ideal of $R=\mathbb{Z}[\alpha ]$, then $R/P$ is a finite field. Complex conjugation preserves $R$, so a nonzero $\beta \in P$ gives the positive integer $\beta \bar{\beta }\in P$. Primality supplies a rational prime $p\in P$. Since $\alpha$ satisfies the monic polynomial $t^2-2at+a^2+b^2$, the ring $R/P$ is a quotient of ${\mathbb{F}}_p[t]/(\bar{F})$ and has at most $p^2$ elements. It is a finite integral domain, hence a field. The proof does not require $R$ to be a principal ideal domain.

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.