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

Prove all of the following statements:

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

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

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

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

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

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

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

Any help would be greatly appreciated, thanks!