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.

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?