# Finding reduced quadratic numbers and principal ideals

Hello :) I want to compute alle reduced quadratic numbers with discriminat $65$. We call a number $\gamma$ reduced if $\gamma>0$ and $-1<\gamma'<0$. We are working in quadratic field extension $\Bbb{Q}(\sqrt{m})$ and intergers in this extension. Thus we know $b^2-4ac=65$ and $\gamma=\frac{-b+\sqrt{65}}{2a}$, thus we must have: $0<-b+\sqrt{65}<2a<b+\sqrt{65}$. But what then? Must we start with a $b$ and find the right $a$ and $c$? Can someone give an example?

Another question: Suppose $\mathfrak{p}$ is a prime ideal above 2 (thus $\mathfrak{p}\cap\Bbb{Z}=p\Bbb{Z}$) in $\Bbb{Q}(\sqrt{65})$. Show that $\mathfrak{p}^2$ a principal ideal is an compute the generator. ..... Here i have no idea how to start. Someone with hints, ideas or solutions?

Thank you :)

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