1. ## Part 1

I have a project I will be working on for the next 2 weeks, and would like some help with the following questions please. I will probably post 3 parts to this over the next week or so. Here is part 1!

Put $\omega:=\frac{-1+ \sqrt{3}i}{2}$, $\mathbb{Z}[\sqrt{3}i]:={a+b\sqrt{3}i | a,b \in \mathbb{Z}}$, $\mathbb{Z}[\omega]:={a+b\omega | a,b \in Z}$ and $S:={a^2 +3b^2 | a,b \in \mathbb{Z}}.$

Prove all of the following statements:

1) $\omega^2=\overline{\omega}=-1-\omega , \delta(\omega)=1$ and $\omega^3=1$

2) (a) $\mathbb{Z}[\sqrt{3}i]$ and $\mathbb{Z}[\omega]$ are subrings of $\mathbb{C}$.

(b) $\mathbb{Z}[\omega]$={ ${\frac{a+b\sqrt{3}i}{2}}$ | $a,b \in \mathbb{Z}, a \equiv b$ $(mod 2)$}

(c) $\mathbb{Z}[\sqrt{3}i] \subseteq \mathbb{Z}[\omega]$

3) (a) Let $a\in \mathbb{Z}[\omega].$ Then $a$ is a unit if and only if $\delta(a) = 1$ and if and only if $a$ is in { $\underline{+}1, \underline{+}\omega, \underline{+} \overline{\omega}$}

(b) For each $a \in \mathbb{Z}[\omega]$ there exists $b \in \mathbb{Z}[\sqrt{3}i]$ such that $a$ is associated to $b$ in $\mathbb{Z}[\omega]$

(c) $S=${ $\delta(a)$ | $a \in \mathbb{Z}[\omega]$}

Any help would be greatly appreciated, thanks!

Put $\omega:=\frac{-1+ \sqrt{3}i}{2}$, $\mathbb{Z}[\sqrt{3}i]:={a+b\sqrt{3}i | a,b \in \mathbb{Z}}$, $\mathbb{Z}[\omega]:={a+b\omega | a,b \in Z}$ and $S:={a^2 +3b^2 | a,b \in \mathbb{Z}}.$
1) $\omega^2=\overline{\omega}=-1-\omega , \delta(\omega)=1$ and $\omega^3=1$
Ok so $\omega ^2$ and $\overline{\omega}$ can be computed fairly easily. I don't think you should have a problem showing the first two equal the third. $\omega^3=1$ also is a fairly straightforward computation as well. I'm probably missing something, but $\delta(\omega)=1$ doesn't make sense to me. If simply means multiply these two terms, then I don't see delta defined anywhere and if delta is a function, then again I don't see any defined function. So besides that I think this one is pretty easy.