Proving $\mathbb Z[i]$ is euclidean domain .

Question

From the definition of euclidean domain , one has to select euclidean function .

Let $\mathbb Z[i]=\{a+bi | a,b\in \mathbb Z,i=\sqrt{-1}\}$

We have to select an euclidean function $f$ , such that

$f$ is from $\mathbb Z[i]-\{0\}$ to $\mathbb N$

my doubt is how to select a good $f$ in-order to prove $\mathbb Z[i]$ is euclidean domain .

Is there any unique method to select or guess euclidean function ?

Are there multiple such $f$'s(euclidean function) to prove $\mathbb Z[i]$ is euclidean domain ?

Answer 1

Define the euclide norm $N(a+bi)=a^2+b^2$ and show that it is actually satisfies your needs

Answer 2

The function $f$ is called evaluation and it's a function $$ f:\mathbb Z[i]\setminus\{0\}\to \mathbb N $$ which has to satisfy to the following properties:

$$ \bullet\;\; f(a)\le f(ab)\;\; \forall a,b\in\mathbb Z[i]\setminus\{0\} $$ $$ \bullet\;\;\forall a,b\in\mathbb Z[i], b\neq0,\;\;\exists\; q,r\in\mathbb Z[i]\\\mbox{s.t.}\\ a=bq+r,\;\;r=0\;\;\mbox{or}\;\;f(r)<f(b)\;\;. $$ The above are the properties to check in order to choose a good function.

The norm $f(a+ib):=a^2+b^2$ verify easily the above properties, hence it's a good choice.