Let $K$ be a number field of degree $n = r_1 + 2r_2$ where $r_1$ is the number of real embeddings and $r_2$ is the number of pairs of complex conjugate embeddings.

Step 1: The Minkowski embedding

Define the Minkowski embedding $\phi: K \to \mathbb{R}^{r_1} \times \mathbb{C}^{r_2}$ by: $\phi(\alpha) = (\sigma_1(\alpha), \ldots, \sigma_{r_1}(\alpha), \tau_1(\alpha), \ldots, \tau_{r_2}(\alpha))$ where $\sigma_i$ are real embeddings and $\tau_j$ are representatives of complex conjugate pairs.

Under this embedding, $\mathcal{O}_K$ maps to a lattice $\Lambda$ in $\mathbb{R}^n$ (identifying $\mathbb{C}^{r_2}$ with $\mathbb{R}^{2r_2}$).

Step 2: Convex body argument

Consider the region $S_t$ defined by: $S_t = \left\{(x_1, \ldots, x_n) \in \mathbb{R}^{r_1} \times \mathbb{C}^{r_2} : \sum_{i=1}^{r_1}|x_i| + 2\sum_{j=1}^{r_2}|x_j| \leq t\right\}$

This region is:

• Convex and symmetric about the origin
• Has volume $V(S_t) = \frac{2^{r_1}(2\pi)^{r_2}}{n!}t^n$

Step 3: Minkowski's lattice theorem

Choose $t$ such that $V(S_t) \geq 2^n \text{vol}(\Lambda)$, where $\text{vol}(\Lambda) = 2^{-r_2}\sqrt{|\Delta_K|}$.

By Minkowski's theorem, $S_t$ contains a nonzero lattice point, i.e., there exists $0 \neq \alpha \in \mathcal{O}_K$ with: $\prod_{i=1}^{r_1}|\sigma_i(\alpha)| \cdot \prod_{j=1}^{r_2}|\tau_j(\alpha)|^2 \leq \left(\frac{n!}{n^n}\right)\left(\frac{4}{\pi}\right)^{r_2}\sqrt{|\Delta_K|}$

The left side equals $|N_{K/\mathbb{Q}}(\alpha)|$, giving a bound on the norm.

Step 4: Bounding ideal norms

Let $\mathfrak{a}$ be any nonzero ideal. Choose $0 \neq \alpha \in \mathfrak{a}$ with norm bounded as above. Then: $N(\mathfrak{a}) \leq N((\alpha)) = |N(\alpha)| \leq M_K$

where $M_K$ is the Minkowski bound.

Step 5: Ideal class representatives

Every ideal class $[\mathfrak{a}]$ contains an ideal $\mathfrak{b}$ with $N(\mathfrak{b}) \leq M_K$.

For each norm $N \leq M_K$, there are only finitely many ideals with that norm (they correspond to divisors of $(N)$ in $\mathcal{O}_K$).

Therefore, there are only finitely many ideals of bounded norm, giving finitely many ideal class representatives.

Conclusion: $h_K = |\text{Cl}(K)| < \infty$.