Theorem 1 (Minkowski's Theorem)   Let $K$ be a number field and let $D_K$ be its discriminant. Let $n=r_1+2r_2$ be the degree of $K$ over $\Rats$ , where $r_1$ and $r_2$ are the number of real and complex embeddings, respectively. The class group of $K$ is denoted by $\Cl(K)$ . In any ideal class $C\in \Cl(K)$ , there exists an ideal $\mathfrak{A}\in C$ such that: $$|{\bf N}(\mathfrak{A})| \leq M_K \sqrt{|D_K|}$$ where ${\bf N}(\mathfrak{A})$ denotes the absolute norm of $\mathfrak{A}$ and $$M_K=\frac{n!}{n^n} \left(\frac{4}{\pi}\right)^{r_2}.$$

Example 1   The discriminants of the quadratic fields $K_2=\Rats(\sqrt{2}),\ K_3=\Rats(\sqrt{3})$ and $K_{13}=\Rats(\sqrt{13})$ are $D_{K_2}=8,\ D_{K_3}=12$ and $D_{K_{13}}=13$ respectively. For all three $n=2=r_1$ and $r_2=0$ . Therefore, the Minkowski's constants are: $$M_{K_i}=\frac{1}{2}\sqrt{|D_{K_i}|},\quad i=2,3,13$$ so in the three cases: $$M_{K_i}\leq \frac{1}{2}\sqrt{13}=1.802\ldots$$ Now, suppose that $C$ is an arbitrary class in $\Cl(K_i)$ . By the theorem, there exists an ideal $\mathfrak{A}$ , representative of $C$ , such that: $$|{\bf N}(\mathfrak{A})|<1.802\ldots <2$$ and therefore ${\bf N}(\mathfrak{A})=1$ . Since the only ideal of norm one is the trivial ideal $\mathcal{O}_{K_i}$ , which is principal, the class $C$ is also the trivial class in $\Cl(K_i)$ . Hence there is only one class in the class group, and the class number is one for the three fields $K_2,\ K_3$ and $K_{13}$ .

Example 2   Let $K=\Rats(\sqrt{17})$ . The discriminant is $D_K=17$ and the Minkowski's bound reads: $$M_K=\frac{1}{2}\sqrt{17}=2.06\ldots$$ Suppose that $C$ is an arbitrary class in $\Cl(K)$ . By the theorem, there exists an ideal $\mathfrak{A}$ , representative of $C$ , such that: $$|{\bf N}(\mathfrak{A})|<2.06\ldots$$ and therefore ${\bf N}(\mathfrak{A})=1$ or $2$ . However, $$2=\frac{-3+\sqrt{17}}{2}\cdot \frac{3+\sqrt{17}}{2}$$ so the ideal $2\mathcal{O}_K$ is split in $K$ and the prime ideals $$\left(\frac{-3+\sqrt{17}}{2} \right), \quad \left( \frac{3+\sqrt{17}}{2}\right)$$ are the only ones of norm $2$ . Since they are principal, the class $C$ is the trivial class, and the class group $\Cl(K)$ is trivial. Hence, the class number of $\Rats(\sqrt{17})$ is one.