Euclidean Domain

definitionring-theoryfield-theory

Definition

We say that an integral domain $R$ is a Euclidean domain if we can define a norm $N:R\to \mathbb{Z}\cup \left\{0\right\}$ such that $N(0)=0$ and for any $a,b\in R$, we have that there exist $q,r\in R$ such that,

$$a=qb+r\text{ with }r=0\text{ or }N(r)<N(b).$$

Thus we can think of a Euclidean domain as an integral domain on which we can define a division algorithm similar to that of $\mathbb{Z}$ or $\mathbb{F}[x]$.

Notes

• $b\mid a$ if and only if $a\in (b)$ if and only if $(a)\subseteq (b)$.

Examples