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Math Help - Norm Question....

  1. #1
    AAM
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    Norm Question....

    Hi guys!

    I've just looked at a past year's number theory questions and one is as follows:

    Let R = Z[Sqrt(-6)] & K = Q[Sqrt(-6)]. Put a = 3+2Sqrt(-6) & b = 7-6Sqrt(-6). By choosing c close to b/a, or otherwise, find c in R such that:

    Normk(b-ac) < Normk(a)

    I have no ideas! :-s

    Any help would be most appreciated! x
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  2. #2
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    Quote Originally Posted by AAM View Post
    Hi guys!

    I've just looked at a past year's number theory questions and one is as follows:

    Let R = Z[Sqrt(-6)] & K = Q[Sqrt(-6)]. Put a = 3+2Sqrt(-6) & b = 7-6Sqrt(-6). By choosing c close to b/a, or otherwise, find c in R such that:

    Normk(b-ac) < Normk(a)

    I have no ideas! :-s

    Any help would be most appreciated! x
    I assume N(x+y\sqrt{-6}) = x^2 + 6y^2. Form \tfrac{b}{a} = p+q\sqrt{-6} where p,q\in \mathbb{Q} by multiplying the numerator and denominator of \tfrac{b}{a} by the conjugate. Next choose p_0,q_0, intergers, so that p_0,q_0 are the closest integers to p,q respectively. Set c = p_0 + q_0 \sqrt{-6} and see that N(b-ac) < N(a).
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  3. #3
    AAM
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    Oh, fantastic! :-D Nice and simple! Just wasn't quite sure what the question was asking for. :-s But I completely get it now!

    Thank you. :-)
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