Principal ideal domain

A principal ideal domain is an integral domain whose every ideal has one generator.

The ascending chain condition supplies existence of factorizations into irreducible elements. The Bezout identity makes every irreducible element prime, which supplies uniqueness up to order and multiplication by units.

Every submodule of a finite-rank free module over a principal ideal domain is free. Induction on rank splits off a generator of the image under one coordinate projection and applies the result to the kernel.

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.