# Proving that the ring is not an euclidean domain

Consider the ring $R= \mathbb Z [(1+\sqrt-19)/2]$. How do I prove it is not an euclidean domain?

-

## 1 Answer

This is a fairly messy (at least as far as I know) proof. The most elementary proof I have seen can be found here

-
+1 I just spent the last 10 minutes reading the proof in the link. I really like this proof; there is clearly a conceptual idea of considering elements of small norm (which is important for our understanding of a Euclidean domain because the Euclidean algorithm tells us that elements of small norm have nice divisibility properties). Moreover, this conceptual idea is converted into a straightforward proof whose details only involve manipulations with complex numbers. – Amitesh Datta Nov 12 '11 at 10:05
@alex youcis : The link which you have attached is no more accessible now.. If you have saved a copy of that or if you have any other link please post it here... Thank You:) – Praphulla Koushik Oct 10 '13 at 16:35