Algebra - Gaussian integers

Question

Let $\mathbb{Z}[i]=\{ a+bi : a,b \in \mathbb{Z}\}$ be the ring of Gaussian integers. Let $x,y \in \mathbb{Z}[i]$ with $y \neq 0$. Show that there exist $q,r \in \mathbb{Z}[i]$ such that $x = yq + r$ and $N(r) < N(y)$, where $N(a+bi)=a^2+b^2$.

I do not have the solution for this nor do i know where to start.

Answer 1

What you are trying to find is like the gcd algorithm, where we want to keep subtracting off a multiple of the 'smaller' number until one of them becomes zero. Of course 'smaller' in $\mathbb{Z}(i)$ has to be defined, and one that works is the distance from $0$, which is sometimes called the norm. Now if I give you two gaussian integers and ask you to do this process, you can imagine all possible multiples of the 'smaller' one, which forms a square grid, and surely you will be able to find a suitable multiple that is close enough to the 'larger' one, which will give you everything you need.

Answer 2

If you are asking for a starting point, you should be aware of this:

1. The fact that you want to prove is an euclidean division for $\Bbb Z[i]$, that is, an operation with similar properties to the division with remainder that you learned at primary school.
2. With this in mind, the quotient $q$ should be "near" the exact quotient $x/y$. Better, it should be the nearest Gaussian integer. Can you guess what is the maximum possible distance $|(x/y)-q|$, provided that $q$ is properly chosen?
3. Use these facts to prove that function $N$ verifies the properties.