If $K$ is a local field, then we will denote $G_K$ the absolute Galois group of $K$, $I_K$ the inertia subgroup of $G_K$ and $P_K$ the wild inertia subgroup of $G_K$, respectively. Now, let $L/K$ be a finite extension. It is clear that $G_L$ is an open subgroup of $G_K$.
Question: What is the relationship between $I_K$ and $I_L$ (resp. $P_K$ and $P_L$)?
It seems that $I_L$ (resp. $P_L$) is not necessarily an open subgroup of $I_K$ (resp. $P_K$). So how to prove the following statement Definition 1.22 in Theory of p-adic Galois Representations by Jean-Marc Fontaine and Yi Ouyang：
Let $\rho:G_K\to {\rm GL}_n(\mathbb{Q}_\ell)$ be a continuous $\ell$-adic representation of $G_K$. Then $\rho(I_K)$ is finite if and only if there exists a finite extension $K'$ of $K$ such that $\rho(I_{K'})$ is trivial. Also, there exists a finite extension $K'/K$ such that $I_K'$ acts unipotently if and only if there exists an open subgroup of $I_K$ which acts unipotently.
Best Answer
$I_K=\mathrm{Gal}(\overline K/K^{\mathrm{unr}})$ and $I_L=\mathrm{Gal}(\overline K/L^{\mathrm{unr}})$. Here, since $K^\mathrm{unr}.L/L$ is an unramified extension, we have a chain of field extensions $K^{\mathrm{unr}}.L/L^{\mathrm{unr}}/K^\mathrm{unr}$. Thus $L^{\mathrm{unr}}/K^\mathrm{unr}$ has finite degree, and in particular $I_L\subset I_K$ has finite index.