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Given a complete valued field $K$ with a discrete value group $\mathbb{Z}$, consider a totally ramified finite Galois extension $L$ of $K$ with its Galois group $G$. Let $O_L$ be the valuation integer ring of $L$, and let $\pi$ be an uniformizer of $L$. For each integer $i\ge 0$, denote $G_i$ for the $i$-th ramification group of $G$, which is the set of automorphisms in $G$ fixing $O_L/(\pi^{i+1})$ pointwise. Let $n$ be the minimum integer such that $G_n={id}$.

Are there any known results about some relations between $n$ and the field extension degree of $L$ over $K$?

Or is there any known result on some special case, for example local fields of characteristic zero?

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    $\begingroup$ There is no bound in terms of $[L:K]$ when working with $p$-adic fields for a fixed prime $p$. Exercise 5 of section 2 of Chapter IV of Serre's book "Local Fields" (the ur-reference on all matters related to the basic ramification theory of local fields) shows that for fixed $p$ and any $n > 0$ if we consider extensions $K$ of $\mathbf{Q}_p$ for which $e(K/\mathbf{Q}_p)$ is large enough (in terms of $n$ and $p$) then $K$ admits a degree-$p$ cyclic extension for which $G_n=G$ and $G_{n+1}=1$. I strongly recommend working through all of the Exercises in that section; very instructive! $\endgroup$
    – nfdc23
    Commented Mar 24, 2016 at 2:28
  • $\begingroup$ Dear nfdc23, I appreciate for the case being $G_n$ nontrivial for sufficiently large $n$. Your example in Serre's 'Local Fields' requires large absolute ramification index $e(K/\mathbb{Q}_p )$. It's interesting. But I also want to find example for a fixed $K$, in special when $K$ absolutely unramified. Fortunately, I've found remarks after proposition13 of section 6 of Chapter 3 of Serre's 'local fields', saying upper and lower bound of different. This gives asymptotic answer for the case I've mentioned. Thanks again! $\endgroup$
    – MiRi_NaE
    Commented Mar 27, 2016 at 7:23

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