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Math Help - In the complex plane

  1. #1
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    Question In the complex plane

    Question 1)
    In the complex plane , there are three cube roots of one. Let zeta be the cube root of one which has positive imaginary part. Show that zeta is a quadratic integer in Q[sqrt(-3)] by writing it in the form (a+bsqrt(-3))/2 , where a and b are rational integers and a and b are either both even or both odd. Then write it in the form m+n*((1+sqrt(-3))/2), where m and n are rational integers.


    I have totally no idea what the questions about.
    Could you please give me detail explaination if you can solve them? Thank you very much.

    Note : I have deleted Question 2 . Since no one reply for it and I have just solved it.
    Last edited by beta12; November 2nd 2006 at 11:38 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by beta12 View Post
    Question 1)
    In the complex plane , there are three cube roots of one. Let zeta be the cube root of one which has positive imaginary part. Show that zeta is a quadratic integer in Q[sqrt(-3)] by writing it in the form (a+bsqrt(-3))/2 , where a and b are rational integers and a and b are either both even or both odd. Then write it in the form m+n*((1+sqrt(-3))/2), where m and n are rational integers.
    The cube root of unity that we want is: z=-1/2 + \sqrt{3}\ i/2=\frac{-1+\sqrt{-3}}{2}.

    Which is in the first form required with a=-1,\ b=1, and the second form with
    m=-1,\ n=1.

    That the root of unity that we require is that given we need to write:

    <br />
1=e^{2n \pi i},\ \ n=0,\ \pm 1,\ ...<br />

    Then the cube roots of unity are:

    <br />
z=e^{2n \pi i/3},\ \ n=0,\ \pm 1,\ ...<br />

    But only three of these are distinct and are given by:

    <br />
z=e^{2n \pi i/3},\ \ n=0,\ \pm 1<br />

    Writing these out in the form a+bi shows that the one given
    above is the one required.

    RonL
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  3. #3
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    Hi Captainblack,

    Thank you for your reply. I will go through it in detail. If I don't understand , I will come back to you. Thank you very much again.
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