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Thread: Part 1

  1. #1
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    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 $\displaystyle \omega:=\frac{-1+ \sqrt{3}i}{2}$, $\displaystyle \mathbb{Z}[\sqrt{3}i]:={a+b\sqrt{3}i | a,b \in \mathbb{Z}}$, $\displaystyle \mathbb{Z}[\omega]:={a+b\omega | a,b \in Z}$ 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!
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  2. #2
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    Quote Originally Posted by shadow_2145 View Post
    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 $\displaystyle \omega^3=1$
    Ok so $\displaystyle \omega ^2$ and $\displaystyle \overline{\omega}$ can be computed fairly easily. I don't think you should have a problem showing the first two equal the third. $\displaystyle \omega^3=1$ also is a fairly straightforward computation as well. I'm probably missing something, but $\displaystyle \delta(\omega)=1$ 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.
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  3. #3
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    I was able to get a start on the first question, but still need help on #2 and #3. Thanks to anyone who could provide help!
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  4. #4
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    Cool, I was able to finish problem # 1. It was easier than I thought, lol. #2 and #3 are still proving to be difficult for me.
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