Let $K$ be a number field of degree $n$ with discriminant $\Delta_K$. Then every ideal class contains an integral ideal $\mathfrak{a}$ with $N(\mathfrak{a}) \leq M_K = \frac{n!}{n^n}\left(\frac{4}{\pi}\right)^{r_2}\sqrt{|\Delta_K|}$ where $r_2$ is the number of pairs of complex embeddings.

Consequence: There are only finitely many ideals of bounded norm, so $h_K = |\text{Cl}(K)| < \infty$.

Let $L/K$ be an abelian extension of number fields. The Artin map induces an isomorphism $\phi: I_K/P_K^L \to \text{Gal}(L/K)$ where $I_K$ is the group of fractional ideals coprime to the conductor, and $P_K^L$ is the subgroup of principal ideals generated by elements $\alpha$ with $(\alpha) \in I_K$ satisfying local norm conditions.

For $L = H_K$ (the Hilbert class field), this gives $\text{Gal}(H_K/K) \cong \text{Cl}(K)$.

For a quadratic field $K = \mathbb{Q}(\sqrt{d})$, the 2-rank of $\text{Cl}(K)$ (dimension of $\text{Cl}(K)/\text{Cl}(K)^2$ as an $\mathbb{F}_2$-vector space) satisfies: $\text{rank}_2(\text{Cl}(K)) = r - 1$ where $r$ is the number of distinct prime divisors of the discriminant $\Delta_K$.

This explains part of the class group structure purely from the discriminant factorization.

For a family of number fields $K_i$ with $[K_i : \mathbb{Q}] = n$ fixed and $|\Delta_{K_i}| \to \infty$: $\lim_{i \to \infty} \frac{\log(h_{K_i} R_{K_i})}{\log\sqrt{|\Delta_{K_i}|}} = 1$

where $R_{K_i}$ is the regulator. This relates the product of class number and regulator to the discriminant asymptotically.

The Stark conjectures predict exact values of $L$-function derivatives at $s = 0$ in terms of units and class numbers. For abelian extensions, these connect:

• Regulators of unit groups
• Class numbers
• Special values of $L$-functions

When proven, they yield explicit class number formulas and constructions of abelian extensions, generalizing Kronecker's Jugendtraum for imaginary quadratic fields.