Inertia group (reviewed)

Let

• $K$ be a $p$-adic field.
• $L$ a finite Galois extension of $K$.
• $\mathcal{O}_K$, $\mathcal{O}_L$ the rings of integers for $K$, $L$,
• $P_K$, $P_L$ the unique maximal ideals of $\mathcal{O}_K$, $\mathcal{O}_L$, and
• $\kappa=\mathcal{O}_K/P_K$, $\lambda=\mathcal{O}_L/P_L$ the residue fields of $K$, $L$.

Then each $\sigma\in \Gal(L/K)$ induces a element of $\Gal(\lambda/\kappa)$. The kernel of the resulting homomorphism \[ \Gal(L/K) \to \Gal(\lambda/\kappa)\] is the inertia group of $L/K$.