Inertia group (reviewed)

Let

• $K$ be a finite Galois extension of $\Q_p$ for some prime $p$,
• $\mathcal{O}_K$ the ring of integers for $K$,
• $P$ the unique maximal ideal of $\mathcal{O}_K$, and
• $k=\mathcal{O}_K/P$.

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