Number theory Q

  1. 1. The problem statement, all variables and given/known data

    Let [itex] \zeta [/itex] be a primative 6-th root of unity. Set [itex] \omega = \zeta i [/itex] where [itex] i^2 = 1[/itex].

    Find a square-free integer [itex]m[/itex] such that [itex] Q [\sqrt{m}] = Q[ \zeta ] [/itex]

    2. Relevant equations

    The minimal polynomial of [itex] \zeta [/itex] is [itex] x^2 - x + 1 [/itex]

    3. The attempt at a solution

    I was intending to use the theorem that:

    Take [itex] p [/itex] to be a prime and [itex] \zeta [/itex] to be a [itex] p [/itex]-th root of unity. if

    [tex]S = \sum_{a =1}^{p-1} \big( \frac{a}{p} \big) \zeta^a [/tex]


    [tex] S^2 = \Big( \frac{-1}{p} \Big) p [/tex].

    This would make [itex]S^2[/itex] an integer. However, 6 is not a prime though. I'm really stumped in what to do. Any help would be greatly appreciated.

    Oh, and by [itex]( \frac{-1}{p} \Big)[/itex], I mean the legendre symbol.
    Last edited: May 12, 2007
  2. jcsd
  3. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    What are the roots of x^2-x+1?
  4. The roots of that polynomial are the primative roots [itex] \zeta [/itex] and [itex]\zeta^5[/itex]. How would I now use this information?
  5. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    No. What are the roots of that polynomial. You're making it too complicated. If I gave you that polynomial in you Freshman calc course, or whatever, you'd be able to write out the roots without thinking. What are the roots? Or better yet, don't write out the roots using THE QUADRATIC FORMULA, just write down a sixth root of unity using elementary complex numbers. HINT: If I asked for a primitive 4th root of unity, would i be acceptable? Or -i? You know that [itex]\exp(2\pi i/n)[/itex] is a primitive n'th root of unity, and that the others are [itex]\exp(2\pi i m/n)[/itex] for m prime to n.
    Last edited: May 13, 2007
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