Chapter Questions
Let $K=\mathbf{Q}(\sqrt{3})$. Use Theorem 10.1 to factorize the following princi$\mathrm{pal}$ ideals in the ring $\mathcal{D}$ of integers of $K$ :$$\langle 2\rangle,(3),(5),\langle 10\rangle,(30) .$$
Factorize the following principal ideals in the ring of integers of $\mathbf{Q}(\sqrt{5})$ :$$(2),\{3\rangle,\langle 5\rangle,\{12\rangle,(25\rangle \text {. }$$
Factorize the following ideals in $\mathbf{Z}(\zeta)$ where $\zeta=e^{2 \pi i / 5}$ :$$(2),(5),(20),(50) \text {. }$$
Compute the volume integral quoted in the proof of Theorem 10.2.
If $K$ is a number field of degree $n$, prowe that$$|\Delta| \geq\left(\frac{\pi}{4}\right)^n\left(\frac{n^n}{n!}\right)^2,$$where $\Delta$ is the discriminant.
Prove that there can exist only finitely many number fields with any given discriminant.
Using the methods of this chapter, compute the class-numbers of fields $Q(\sqrt{d})$ for $-20 \leq d \leq 20$.