Gaussian Integers form Euclidean Domain

Theorem
Let $\left({\Z \left[{i}\right], +, \times}\right)$ be the integral domain of Gaussian Integers.

Let $\nu: \Z \left[{i}\right] \to \R$ be the real-valued function defined as:
 * $\forall a \in \Z \left[{i}\right]: \nu \left({a}\right) = \left \vert{a}\right \vert^2$

where $\left \vert{a}\right \vert$ is the (complex) modulus of $a$.

Then $\nu$ is a Euclidean valuation on $\Z \left[{i}\right]$.

Hence $\left({\Z \left[{i}\right], +, \times}\right)$ with $\nu: \Z \left[{i}\right] \to \Z$ forms a euclidean domain.