# Euclidean Domain/Examples

## Examples of Euclidean Domains

### Integers are Euclidean Domain

The integers $\Z$ with the mapping $\nu: \Z \to \Z$ defined as:

$\forall x \in \Z: \map \nu x = \size x$

form a Euclidean domain.

### Polynomial Forms over Field is Euclidean Domain

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $X$ be transcendental in $F$.

Let $F \sqbrk X$ be the ring of polynomial forms in $X$ over $F$.

Then $F \sqbrk X$ is a Euclidean domain.

### Gaussian Integers form Euclidean Domain

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.