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Thread: A couple quadratic field questions

  1. #1
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    A couple quadratic field questions

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


    2) Consider $\displaystyle \mathbf{Q}\left(\sqrt{-1}\right)$. Write an equation relating $\displaystyle N(\alpha)$ to $\displaystyle |\alpha|$ (the natural absolute value defined for complex numbers). For which $\displaystyle \mathbf{Q}\left(\sqrt{d}\right)$ 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 $\displaystyle \mathbf{Q}\left(\sqrt{d}\right)$ has elements $\displaystyle \alpha$ with negative norm $\displaystyle N(\alpha)$. Assume d is not a perfect square.
    I am confused because a norm is $\displaystyle N: \mathbb{Q}(\sqrt{d}) \to \mathbb{N}$. Thus, it is never negative.

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

    For the second part I think the norm you are referring to is $\displaystyle N(x+iy) = x^2+y^2$. Thus, $\displaystyle N(\alpha) = |\alpha|^2$.
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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 $\displaystyle N(a+b\sqrt{d}) = a^2 - db^2$.
    Yes, the book I'm working out of defines a norm of $\displaystyle \alpha$ to be the number $\displaystyle N(\alpha)=\alpha\bar{\alpha}$ where $\displaystyle \bar{\alpha}$ is the conjugate of $\displaystyle \alpha$.
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