Euclidean algorithm in a Euclidean domain

Summary

Let $R$ be a Euclidean domain with Euclidean function $\delta$ and let $a, b \in R$. The Euclidean algorithm sets $r_0 = a$ and $r_1 = b$. While $r_k \neq 0$, it divides $r_{k-1}$ by $r_k$ and lets $r_{k+1}$ be the remainder. The values $\delta(r_k)$ decrease strictly, so the algorithm stops with some $r_{n+1} = 0$. The last nonzero remainder $r_n$ is a greatest common divisor of $a$ and $b$, if $b \neq 0$.