Given a local field K and its finite galois extension L (we have non-archimedean valuation $v\_{K}$ and $v\_{L}$). We define $\Gamma=Gal(L/K)$, and for $i\in \mathbb{Z}$, we can define $\Gamma\_{i}$ (the $\Gamma\_{0}$ is the inertia group). I want to know if $\Gamma\_{i}$ is nontrivial, whether it strictly contains $\Gamma\_{i+1}$ or not? 




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Here is the definition of $\Gamma\_{i}$$:=$$\{\gamma\in\Gamma|v\_{L}(x\gamma-x)\geq i+1 ,\forall x\in R\_{L}\}$, $R\_{L}$ is the valuation ring



Is the filtration of inertia group a strict one?
No, the inclusion of $\Gamma\_{i+1}$ in $\Gamma\_i$ is often an equality. The simplest example is $K=\mathbf{Q}\_2$, $L=\mathbf{Q}\_2(\sqrt2)$. Here $\Gamma=\Gamma\_0=\Gamma\_1=\Gamma\_2$, and $\Gamma\_3$ is trivial. 


It is possible to find a $K$ and a degree-$p$ cyclic extension $L$ of $K$ (where $p$ is the residual characteristic) such that the length of the filtration is any preassigned number $\not\equiv0\pmod p$.


