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To give the context I am currently studying Gaussian Integers, and I have of course studied rings. The full question is:

Given $\rho = \exp(\frac{2\pi i}{3})$, show that the ring $R = \mathbb{Z}[\rho]$ together with $N(z) = |z|^2$, for $z \in R$, is a Euclidean domain.

My bigest problem so far is that I am confused about $\mathbb{Z}[\rho]$: I believe it refers to the polynomial ring of variable $\rho$ where $i$ is the imaginary number. However it doesn't really make sens since $\rho$ is not a variable but a constant. Therefore I thought it might refer to the ring of integers modulo $\rho$, but if it is so I can't see clearly what it represents.

Furthermore, I know there is no general way to prove something is a Euclidean domain, so if we are to do it case by case I would like to be positive how to begin. The only thing that comes to my mind so far is to prove that $\forall \rho_1,\rho_2 \in R$, $\exists q,r \in R$ such that $\rho_1 = q\rho_2 + r$ where either $r = 0$ or $N(r) < N(\rho_2)$.

I've already looked up for something similar or just a general case on the internet and on Mathstack but all I could I could find was How do I prove whether something is a Euclidean domain? which is not relevant in this situation.

I hope you can help me with the basics of this problem so I can move on a little! I only hope for hints.

user26857
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Rhaena
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    $\mathbb Z[\rho]$ is the ring $\mathbb Z$ with $\rho$ adjoined; the smallest ring containing $\mathbb Z$ and $\rho$ – J. W. Tanner Nov 30 '20 at 17:16
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    elements of $\mathbb Z[\rho]$ are $a+b\rho$, where $a,b\in\mathbb Z$; these are the Eisenstein integers – J. W. Tanner Nov 30 '20 at 17:18
  • I had not seen it from this perspective so thanks @J.W.Tanner ! Although I had not seen this property of smallest ring containing $\mathbb{Z}$ and $\rho$, it makes total sens – Rhaena Nov 30 '20 at 17:23
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    Hint. Use @J.W.Tanner 's comments to draw a picture showing the elements of that ring as points in the complex plane. Then find the Eisenstein integer closest to $\rho_1/\rho_2$. – Ethan Bolker Nov 30 '20 at 17:24
  • Thank you @Ethan, I was indeed trying to draw this on my paper and it took me some time to find it, but I think I have! However this drawing is not enough to find $\frac{\rho_1}{\rho_2}$, would it work with the Euclidean algorithm ? Considering rational coefficients I mean – Rhaena Nov 30 '20 at 17:40
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    The Eisenstein integer nearest the quotient is the $q$ in the Euclidean algorithm. – Ethan Bolker Nov 30 '20 at 18:26
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    given a ratio of Eisenstein integers, you rationalize the denominator, with $(x + y \rho) (x + y \rho^2) = x^2 - xy + y^2 $ The numerator is multiplied out, using $\rho^2 = -1 - \rho$ You get the ratio in the form $c + d \rho$ with $c,d \in \mathbb Q$ Here I have displayed the norm that they wrote $N(z) = |z|^2$ – Will Jagy Nov 30 '20 at 18:35
  • Indeed! thank you all for your help, I think I can finish from there! – Rhaena Nov 30 '20 at 18:39

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