1. ## 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.

2. Originally Posted by Pn0yS0ld13r
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, $\displaystyle N(\alpha) = |\alpha|^2$.

3. Originally Posted by ThePerfectHacker
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 $\displaystyle \alpha$.