## How to prove that Z[cis(2pi/3)] is euclidean ? [closed]

Hi all, Z[cis(2pi/3)]:={a+bw|a,b are integers,w=cos(2pi/3)+isin(2pi/3)}.

I want to show that Z[w] is euclidean , I tried doing it in similar fashion to the way we show (at least the proofe I know) that Z[i] (the gaussian integers) is euclidean. (the reason for the problem is that cos(2pi/3) - isin(2pi/3) is not in Z[w] (moreover it is not in Q[w])).

Help is apprecciated.

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 I don't get part of the the statement you made - if $w=\cos(2\pi/3)+i\sin(2\pi/3)\in\mathbb{Z}[w]$ then $w^2=\cos(2\pi/3)-i\sin(2\pi/3)$ is in $\mathbb{Z}[w]$ too! – Somnath Basu May 17 2011 at 17:24 I think this is a question that you can solve on your own. Please ask the internet only when you are stuck on more difficult questions (and at this level, your questions are more appropriate at math.stackexchange.com ). – S. Carnahan♦ May 17 2011 at 17:43