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Math Help - Ideal Question....

  1. #1
    AAM
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    Ideal Question....

    Dealing with Z[Sqrt(-13)], let J = (7, -1+Sqrt(-13) ). ie - J is the ideal generate by 7 & -1+Sqrt(-13)

    Find an integer N s.t. J^2 = (49, n+Sqrt(-13) )

    Many thanks in advance. :-) x
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  2. #2
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    Quote Originally Posted by AAM View Post
    Dealing with Z[Sqrt(-13)], let J = (7, -1+Sqrt(-13) ). ie - J is the ideal generate by 7 & -1+Sqrt(-13)

    Find an integer N s.t. J^2 = (49, n+Sqrt(-13) )
    Not being an algebraist, I'm not 100% sure of this, but it looks to me as though J^2 should contain the elements 7^2=49, (-1+\sqrt{-13})^2 = -12-2\sqrt{-13} and 7(-1+\sqrt{-13}) = -7+7\sqrt{-13}. Therefore it will also contain 49 + 3(-12-2\sqrt{-13}) + (-7+7\sqrt{-13}) = 6 + \sqrt{-13}. So I would go for n=6.
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  3. #3
    AAM
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    Awesome - thank you! :-)

    Is there a more efficient way of finding n, or do you just have to "play around with numbers" as you have done? :-)
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