Euclidean domain

Definition

A Euclidean domain is an integral domain $R$ with a function $\delta \colon R \setminus \{0\} \to \mathbb{N}$ with this property. For all $a, b \in R$ with $b \neq 0$, there are $q, r \in R$ such that $a = b q + r$, and $r = 0$ or $\delta(r) < \delta(b)$. The function $\delta$ is a Euclidean function of $R$.

Remarks

The absolute value is a Euclidean function on $\mathbb{Z}$. The degree is a Euclidean function on $K[x]$ for a field $K$. Some authors also require $\delta(a) \le \delta(a b)$ for all nonzero $a, b$. This changes the class of Euclidean domains only in appearance, because a ring with a Euclidean function has one with this extra property. Every Euclidean domain is a principal ideal domain.

Explanations

L1 human

In a Euclidean domain, you can always divide with a remainder that is smaller than the divisor. The function $\delta$ measures what smaller means. This is all you need to run the Euclidean algorithm.