Let $R$ be a Dedekind domain. Every nonzero ideal $I$ of $R$ can be written uniquely as a product of prime ideals: $I = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_r^{e_r}$ where $\mathfrak{p}_i$ are distinct prime ideals and $e_i \geq 1$. The factorization is unique up to reordering of factors.

Moreover, every nonzero fractional ideal has a unique factorization $I = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_r^{e_r}$ where now $e_i \in \mathbb{Z}$ (possibly negative).

For an integral domain $R$, the following are equivalent:

1. $R$ is a Dedekind domain
2. Every nonzero fractional ideal of $R$ is invertible
3. $R$ is Noetherian, integrally closed, and every nonzero prime ideal is maximal
4. $R$ is Noetherian and the localization $R_\mathfrak{p}$ at each prime $\mathfrak{p}$ is a discrete valuation ring (DVR)

The multiplicative structure of fractional ideals gives an exact sequence: $1 \to \mathcal{O}_K^* \to K^* \to \text{Frac}(\mathcal{O}_K) \to \text{Cl}(K) \to 0$

where $\text{Frac}(\mathcal{O}_K)$ is the group of fractional ideals and $\text{Cl}(K)$ is the ideal class group. This sequence encodes:

• Elements vs. principal ideals: $K^* \to \text{Frac}(\mathcal{O}_K)$ sends $\alpha \mapsto (\alpha)$
• Class group as quotient: $\text{Cl}(K) = \text{Frac}(\mathcal{O}_K) / K^*$

Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ satisfies monic $f(x) \in \mathbb{Z}[x]$. For prime $p$ not dividing $[\mathcal{O}_K : \mathbb{Z}[\alpha]]$, if $f(x) \equiv \prod_{i=1}^g \bar{f}_i(x)^{e_i} \pmod{p}$ where $\bar{f}_i$ are distinct irreducibles in $\mathbb{F}_p[x]$, then $(p) = \prod_{i=1}^g \mathfrak{p}_i^{e_i}, \quad \mathfrak{p}_i = (p, f_i(\alpha))$

where $f_i$ is a lift of $\bar{f}_i$ to $\mathbb{Z}[x]$, and the inertia degree of $\mathfrak{p}_i$ is $\deg(f_i)$.