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Math Help - A couple quadratic field questions

  1. #1
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    A couple quadratic field questions

    1) Find for which integers d the field \mathbf{Q}\left(\sqrt{d}\right) has elements \alpha with negative norm N(\alpha). Assume d is not a perfect square.


    2) Consider \mathbf{Q}\left(\sqrt{-1}\right). Write an equation relating N(\alpha) to |\alpha| (the natural absolute value defined for complex numbers). For which \mathbf{Q}\left(\sqrt{d}\right) is this formula correct?

    How do you approach these questions?

    Thanks.
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  2. #2
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    Quote Originally Posted by Pn0yS0ld13r View Post
    1) Find for which integers d the field \mathbf{Q}\left(\sqrt{d}\right) has elements \alpha with negative norm N(\alpha). Assume d is not a perfect square.
    I am confused because a norm is N: \mathbb{Q}(\sqrt{d}) \to \mathbb{N}. Thus, it is never negative.

    Thus, perhaps you are defining the norm to be N(a+b\sqrt{d}) = a^2 - db^2. If d<0 then clearly it is always non-negative. However, if d>0 then N(0+b\sqrt{d})<0 where b\not = 0.

    For the second part I think the norm you are referring to is N(x+iy) = x^2+y^2. Thus, N(\alpha) = |\alpha|^2.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Thus, perhaps you are defining the norm to be N(a+b\sqrt{d}) = a^2 - db^2.
    Yes, the book I'm working out of defines a norm of \alpha to be the number N(\alpha)=\alpha\bar{\alpha} where \bar{\alpha} is the conjugate of \alpha.
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