Gaussian Integers form Principal Ideal Domain

Theorem
The ring of Gaussian integers:
 * $\struct {\Z \sqbrk i, +, \times}$

forms a principal ideal domain.

Proof
From Gaussian Integers form Integral Domain, we have that $\struct {\Z \sqbrk i, +, \times}$ is an integral domain.

Let $a, d \in \Z \sqbrk i$ such that $d \ne 0$.

Suppose $\cmod a \ge \cmod d$.

Reference to an Argand diagram shows that one of:
 * $a + d, a - d, a + i d, a - i d$

is closer to the origin than $a$ is.

So it is possible to subtract Gaussian integer multiples of $d$ from $a$ until the square of the modulus of the remainder drops below $\cmod d^2$.

That remainder can only take integer values.

Thus a Division Theorem result follows:


 * $\exists q, r \in \Z \sqbrk i: a = q d + r$

where $\cmod r < \cmod d$.

Let $J$ be an arbitrary non-null ideal of $\Z \sqbrk i$.

Let $d$ be an element of minimum modulus in $J$.

Then the Division Theorem can be used to prove that $J = \ideal d$.