A Ring $R$ is called euclidean if a map $f:R\backslash {0} \rightarrow \mathbb{N}$ exists with the following properties: For two elements $a,b \in R$ with $b\neq 0$ there exist $q,r\in R$ with:
(i) $a=qb+r$ and
(ii) $r=0$ or $f\left(r \right)<f\left(b\right)$.
The Gaussian Integers are a Ring $G:= \{a+bi:a,b \in\mathbb{Z}\}$. I want to show that $G$ is an euclidean Ring. Because $G$ is obviously an Integral domain, I would be able to show that $G$ is a Principal ideal domain.
My problem is with the second part (ii). I saw that such a mapping can be defined with $a+bi \mapsto a^2 + b^2$. Here is what I don't get: In our case the $b$ in (ii) is i, so I want to find a $f$ with $f\left(a \right)<f\left(i\right)$ for the Gaussian Integers right? How is $a+bi \mapsto a^2 + b^2$ helping here? I am sorry if the notation is a bit odd. For the Gaussian integers I get (to put them into the frame like in (i)) that $a=q$, $b=i$ and $r=a$ right?
Appreciate your help, KingDingeling
Best Answer
You seem to be a bit confused. $b$ could be any element of $G$ except $0$. You have already found a good choice for $f$ so the only part left is, given arbitrary $a, b, \in G$ with $b \neq 0$, to find $q, r \in G$ satisfying (i) and (ii).