Can we do long division in $\mathbb Z[\sqrt n]$, where $n$ is square free? Can we do long division of elements $a + b\sqrt n$ and $c + d\sqrt n$ in $\mathbb Z[\sqrt n]$? I gave it a shot, but wasn't sure how to proceed... I'd like to use the gcd algorithm to show that two ideals <a,b> and <c> in that set are equal.
Also, the norms of those elements can tell us things about the elements. Like, if the norms are prime then the elements are irreducible? So could finding the gcd of norms shed light on the equality of these ideals? My text book gives one example in Z of using gcd(a,b) = c to conclude that <a,b> = <c>, but doesn't give any examples of this failing, or of its application in complex extensions.
Thanks,
a.

In general, $\mathbb Z[\sqrt n]$ need not be an euclidean ring.
