Let $L/K$ be an extension of number fields and $\mathfrak{p}$ a prime of $\mathcal{O}_K$. The ideal $\mathfrak{p}\mathcal{O}_L$ factors uniquely in $\mathcal{O}_L$ as: $\mathfrak{p}\mathcal{O}_L = \mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_g^{e_g}$

where $\mathfrak{P}_i$ are distinct primes of $\mathcal{O}_L$ lying above $\mathfrak{p}$. The exponent $e_i$ is the ramification index of $\mathfrak{P}_i$ over $\mathfrak{p}$.

The prime $\mathfrak{p}$ ramifies in $L$ if some $e_i > 1$. Otherwise, $\mathfrak{p}$ is unramified.

For a prime $\mathfrak{P}$ of $\mathcal{O}_L$ lying above $\mathfrak{p}$ of $\mathcal{O}_K$, the inertia degree (or residue class degree) is: $f(\mathfrak{P}|\mathfrak{p}) = [(\mathcal{O}_L/\mathfrak{P}) : (\mathcal{O}_K/\mathfrak{p})]$

the degree of the residue field extension. The fundamental relation is: $\sum_{i=1}^g e_i f_i = [L : K]$

where $e_i = e(\mathfrak{P}_i|\mathfrak{p})$ and $f_i = f(\mathfrak{P}_i|\mathfrak{p})$.

A prime $\mathfrak{p}$ ramifies in $L/K$ if and only if $\mathfrak{p}$ divides the different $\mathfrak{D}_{L/K}$.

The discriminant $\Delta_{L/K}$ is the ideal generated by $\det(\text{Tr}_{L/K}(\omega_i\omega_j))$ for any integral basis $\{\omega_1, \ldots, \omega_n\}$ of $\mathcal{O}_L$.

The formula $\mathfrak{D}_{L/K} = \prod_{\mathfrak{p}} \mathfrak{p}^{d_\mathfrak{p}}$ characterizes ramification: $d_\mathfrak{p} > 0$ precisely when $\mathfrak{p}$ ramifies.

For a prime $\mathfrak{P}$ of $L$ above $\mathfrak{p}$ of $K$ in a Galois extension $L/K$, the decomposition group is: $D_\mathfrak{P} = \{\sigma \in \text{Gal}(L/K) : \sigma(\mathfrak{P}) = \mathfrak{P}\}$

This subgroup encodes how Galois automorphisms interact with $\mathfrak{P}$. The quotient by the inertia group $I_\mathfrak{P}$ gives the Galois group of the residue field extension.