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Math Help - 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 \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!
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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 \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
    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.
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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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