Gaussian Integers form Euclidean Domain

Theorem
Let $\struct {\Z \sqbrk i, +, \times}$ be the integral domain of Gaussian Integers.

Let $\nu: \Z \sqbrk i \to \R$ be the real-valued function defined as:
 * $\forall a \in \Z \sqbrk i: \map \nu a = \cmod a^2$

where $\cmod a$ is the (complex) modulus of $a$.

Then $\nu$ is a Euclidean valuation on $\Z \sqbrk i$.

Hence $\struct {\Z \sqbrk i, +, \times}$ with $\nu: \Z \sqbrk i \to \Z$ forms a Euclidean domain.