Euclidean Domain

This is a Integral Domain with a Divison Algorithm. That is:

Formal Definition

• With $\mathcal{D}$ as an Integral Domain
• $\mathcal{D}$ is a euclidean domain if $\forall a,b\in \mathcal{D}$, $\exists$ Norm on $\mathcal{D}$ $N:\mathcal{D}\to {\mathbb{Z}}^{+}$ s.t:
  1. $\forall a,b\in \mathbb{R},b\ne 0,\exists q,r\in \mathcal{D}$ s.t $a=bq+r$
  2. $a,b\ne 0⟹N(a)\le N(ab)$

Concepts

• Euclidean Metric for Complex Numbers