Let $L/K$ be a Galois extension with group $G$. For a prime $\mathfrak{p}$ of $K$:

1. All primes $\mathfrak{P}_1, \ldots, \mathfrak{P}_g$ above $\mathfrak{p}$ are conjugate under $G$
2. The decomposition groups $D_{\mathfrak{P}_i}$ are conjugate subgroups of $G$
3. If $L/K$ is abelian, all $D_{\mathfrak{P}_i}$ coincide as subgroups of $G$
4. The ramification index $e$, inertia degree $f$, and number $g$ satisfy $efg = [L : K]$

Moreover, $|D_\mathfrak{P}| = ef$ and $|I_\mathfrak{P}| = e$ for any prime $\mathfrak{P}$ above $\mathfrak{p}$.

A prime $\mathfrak{p}$ of $K$ ramifies in $L/K$ if and only if $\mathfrak{p}$ divides the discriminant $\Delta_{L/K}$.

Explicitly: $\mathfrak{p}^{a_\mathfrak{p}} || \Delta_{L/K}$ where $a_\mathfrak{p} = \sum_{\mathfrak{P} | \mathfrak{p}} (e_\mathfrak{P} - 1 + \delta_\mathfrak{P})$ and $\delta_\mathfrak{P}$ measures wild ramification.

This gives a finite set of ramified primes: $\text{Ram}(L/K) = \{\mathfrak{p} : \mathfrak{p} | \Delta_{L/K}\}$.

For an unramified prime $\mathfrak{p}$ in a Galois extension $L/K$, the Frobenius element $\text{Frob}_\mathfrak{P} \in D_\mathfrak{P}/I_\mathfrak{P}$ is characterized by: $\text{Frob}_\mathfrak{P}(\alpha) \equiv \alpha^{N(\mathfrak{p})} \pmod{\mathfrak{P}}$

for all $\alpha \in \mathcal{O}_L$. When $L/K$ is abelian, $\text{Frob}_\mathfrak{P}$ is independent of the choice of $\mathfrak{P}$ above $\mathfrak{p}$ and lies in $\text{Gal}(L/K)$.

The Chebotarev density theorem: conjugacy classes of Frobenius elements are equidistributed among primes.

If $L_1/K$ and $L_2/K$ are extensions with coprime ramification indices at all primes, then the compositum $L_1 L_2$ has ramification indices: $e_{L_1 L_2/K}(\mathfrak{P}) = \text{lcm}(e_{L_1/K}(\mathfrak{P}_1), e_{L_2/K}(\mathfrak{P}_2))$

When ramification indices are coprime, this equals their product, giving mild ramification in composita.

For a Galois extension $L/K$ with group $G$, the quotient $I_\mathfrak{P}/I_\mathfrak{P}^{(1)}$ (where $I_\mathfrak{P}^{(1)}$ is the first higher ramification group) is cyclic of order prime to the residue characteristic.

This structure theorem for inertia groups is fundamental to understanding tame vs. wild ramification.