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Let $\zeta$ be the cube root of 1 given by $\zeta=\frac{-1}{2}+i\frac{\sqrt{3}}{2}$ and let $\mathbb{Z}[\zeta]=\{a+\zeta b: a, b\in \mathbb{Z}\}$, called the "Eisenstein integers".

How prove the following exercises?

(a) Every element of $\mathbb{Z}[\zeta]$ can be uniquenly written in the form $a+\zeta b$ for some $a, b\in\mathbb{Z}$.

(b) Let $N(a+\zeta b)=(a+\zeta b)(a+\overline{\zeta} b)=a^2-ab+b^2$. Show that the units of $\mathbb{Z}[\zeta]$ are $\{\pm 1,\pm\zeta, \pm \zeta^2 \}$.

(c) Show $\mathbb{Z}[\zeta]$ is an Euclidean domain with size function $N$.

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    The proof of uniqueness almost writes itself. Suppose that $a+b\zeta=c+d \zeta$. Then $a-c=(d-b)\zeta$. But $\dots$.2012-06-29
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    André. I already did Part 1. How get the part two an three?2012-06-29
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    Dear Andres, Then you should say that you have done Part 1 in the body of the question, so that André does not spend his time telling you what you already know. For (b), can you show that those six elements are in fact units? That's a start.2012-06-29
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    Inthe part 1. I suposse that $a+\zeta b=a'+\zeta b'$ and prove that $a=a'$ and $b=b'$2012-06-29

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