I have some problems with the topic "fractional ideals". I have two questions:
- Compute a generator $\alpha$ of the fractional ideal $\Bbb{Z}+\Bbb{Z}(\phi^3(5+\sqrt{31}))$, thus find $\alpha\in\Bbb{Q}(\sqrt{31})$ such that $\Bbb{Z}+\Bbb{Z}(\phi^3(5+\sqrt{31}))=\Bbb{Z}[\sqrt{31}]\alpha$.
- Suppose $\gamma$ irrational real number. Prove that $\phi(\gamma)(\Bbb{Z}+\Bbb{Z}\gamma)=\Bbb{Z}+\Bbb{Z}\phi(\gamma)$. Suppose also that $\Bbb{Z}+\Bbb{Z}\gamma$ is a fractional ideal in $\Bbb{Z}[\sqrt{31}]$ generated by $\alpha$. Show that $\Bbb{Z}+\Bbb{Z}\phi(\gamma)$ is generated by $\phi(\gamma)\alpha$
Here we notated $\phi$ as a function given by $\phi(x)=\frac{1}{x-\left\lfloor x\right\rfloor}$ with $\left\lfloor x\right\rfloor$ the floor function. I'm learing for my exam and i hope someone can help me. Thank you :)
For the second item, note that $u\mathbb Z + v\mathbb Z = w\mathbb Z + x\mathbb Z$ holds if $\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\left(\begin{matrix}w\\x\end{matrix}\right)$ for some matrix $\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\in\operatorname{SL}_2(\mathbb Z)$ because that matrix allows you to express the respective generators.