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

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

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    +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.2011-11-12
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    @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:)2013-10-10
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    Here is a link to a file of the same name, I'm not sure it's the same one: http://www.maths.qmul.ac.uk/~raw/MTH5100/PIDnotED.pdf2016-06-04