Euclidean Domain/Examples
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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.