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Generalized Gaussian integers

  1. Sep 13, 2009 #1
    1. The problem statement, all variables and given/known data
    If [tex]\omega[/tex] is and nth root of unity, define Z[[tex]\omega[/tex]], the set of generalized Gaussian integers to be the set of all complex numbers of the form
    m[tex]_{0}[/tex]+m[tex]_{1}\omega[/tex]+m[tex]_{2}\omega^{2}[/tex]+...+m[tex]_{n-1}\omega^{n-1}[/tex]
    where n and m[tex]_{i}[/tex] are integers.
    Prove that the products of generalized Gaussian integers are generalized Gaussian integers.


    2. Relevant equations



    3. The attempt at a solution
    I'm not sure how to start this, so a hint or two would be greatly appreciated.
     
  2. jcsd
  3. Sep 13, 2009 #2

    Dick

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    As a first step, you want to prove that w^a*w^b is also an nth root of unity for any integers a and b. Can you handle that?
     
  4. Sep 13, 2009 #3
    w^a*w^b= w^(a+b)
    but what exactly do I need to show to prove it is an nth root of unity?
     
  5. Sep 13, 2009 #4

    Dick

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    You need to show that the nth power of w^(a+b) is one. That's what a nth root of unity is.
     
  6. Sep 13, 2009 #5
    Okay, I got that now, but am unsure of the next step and how it relates to the generalized Gaussian integers.
     
  7. Sep 13, 2009 #6

    Dick

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    Take the product of two of those generalized Gaussian integers. Distribute the product. What kind of terms do you get?
     
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