# Euclidean Domains

• January 20th 2009, 10:16 PM
dori1123
Euclidean Domains
Let $R$ be a Euclidean Domain. Let $m$ be the minimum integer in the set of norms of nonzero elements of $R$. Prove that every nonzero element of norm $m$ is a unit in $R$ . Deduce that a nonzero element of norm zero (if such an element exists) is a unit.
• January 21st 2009, 08:40 AM
ThePerfectHacker
Quote:

Originally Posted by dori1123
Let $R$ be a Euclidean Domain. Let $m$ be the minimum integer in the set of norms of nonzero elements of $R$. Prove that every nonzero element of norm $m$ is a unit in $R$ . Deduce that a nonzero element of norm zero (if such an element exists) is a unit.

A norm is a function $n: R^{\times}\to \mathbb{N}$ such that for any $a,b\in R$, $b\not = 0$ we can write $a = qb + r$ where: $r\not = 0$ and $n(r) < n(b)$, or $r=0$. (There might be an additional property to the norm function but not all books define it).

Say that $n(x) = m$ where $m = \min \{ n(r) : r\in \mathbb{R}^{\times}\}$. This means we can write, $1 = qx + r$ where $r=0$ or else $n(r) < n(x)$. But $n(x)$ is minimal and so $r=0$. Thus, $1=qx$ which means that $x|1$. Thus, $x$ is a unit by definition. It follows that if $n(x) = 0$ then since $x$ has the smallest possible norm it means it has to be a unit.