TITLE: How do I prove whether something is a Euclidean domain?
QUESTION [6 upvotes]: Is there a "special formula" one can follow to prove whether something is a Euclidean domain or not? I've been looking around, but I haven't seemed to be able to find one, so I was wondering whether I was blind, or there just isn't one.
I have $\mathbb Z[\sqrt-3]=\{x+y\sqrt-3|x,y\in\mathbb Z\}$. I think I can remember having read somewhere that it's a Euclidean domain, but I'm not sure. I also don't know how to prove it. Any hints?

REPLY [5 votes]: To prove something is or is not a Euclidean domain, it seems useful to use the following chart:
Fields $\subset$ Euclidean Domain $\subset$ Principal Ideal Domain $\subset$ Unique Factorization Domain $\subset$ Domain
In particular, to prove something is a Euclidean domain, you may prove either it is a field (only if it actually is a field), or you may prove it is a Euclidean domain directly (See below for details).
To prove something is not a euclidean domain, you may prove that it is not one of the latter ones: i.e., prove there exists an ideal that is not principal, a factorization that is not unique, or zero divisors.
To prove something is a euclidean domain, by and large you must prove the existence of a division algorithm using the standard definition that $\forall x,y \in D \exists q,r$ satisfying $$ x = qy + r$$ and $\mathcal{N}(r) < \mathcal{N}(y)$, where $\mathcal{N}$ is a norm on your domain $D$.