Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Two fractional ideals $I$ and $J$ are equivalent if there exist nonzero $\alpha, \beta \in K$ such that $\alpha I = \beta J$.

The set of equivalence classes forms a group under ideal multiplication, called the ideal class group or simply class group, denoted $\text{Cl}(K)$ or $\text{Cl}_K$.

The identity element is the class of principal ideals, and the order $h_K = |\text{Cl}(K)|$ is the class number of $K$.

A set of integral ideals $\{\mathfrak{a}_1, \ldots, \mathfrak{a}_h\}$ is a complete set of representatives for $\text{Cl}(K)$ if every ideal class contains exactly one $\mathfrak{a}_i$.

By the Minkowski bound, representatives can be chosen with bounded norm: $N(\mathfrak{a}_i) \leq M_K = \frac{n!}{n^n}\left(\frac{4}{\pi}\right)^{r_2}\sqrt{|\Delta_K|}$ where $n = [K : \mathbb{Q}]$ and $r_2$ is the number of complex conjugate pairs of embeddings.

The narrow class group $\text{Cl}^+(K)$ refines the class group by requiring equivalence via totally positive elements: $I \sim J$ if $\alpha I = \beta J$ for $\alpha, \beta$ totally positive (positive at all real embeddings).

The narrow class number $h^+_K = |\text{Cl}^+(K)|$ satisfies $h_K | h^+_K$ and $h^+_K / h_K \in \{1, 2^{r_1}\}$ where $r_1$ is the number of real embeddings.