= Solution
For a finite <Galois extension> of local fields, let $G=\operatorname{Gal}(L/K)$, $G_{-1}=G$, and, for integers $i\geq0$, define the lower <ramification groups> by
$$
G_i=\{\sigma\in G:v_L(\sigma(a)-a)\geq i+1\text{ for all }a\in\mathcal O_L\}.
$$
In particular, $G_0$ is the <inertia group>. The identity satisfies every bound because $v_L(0)=\infty$.
If the extension is <totally ramified>, its <residue fields> agree. Choose representatives in $\mathcal O_K$ of their common <residue field>. Every $a\in\mathcal O_L$ has a convergent <uniformiser> expansion $a=\sum_{j\geq0}c_j\pi_L^j$ with these representatives $c_j$, all fixed by $G$. For $j\geq1$,
$$
\sigma(\pi_L)^j-\pi_L^j=(\sigma(\pi_L)-\pi_L)\sum_{r=0}^{j-1}\sigma(\pi_L)^r\pi_L^{j-1-r}.
$$
Each summand in the second factor has <valuation> $j-1$. Thus a bound $v_L(\sigma(\pi_L)-\pi_L)\geq i+1$ implies the same bound on $\sigma(a)-a$, by convergence and the <valuation> inequality. Necessity follows by testing $a=\pi_L$. This proves the <uniformizer criterion for lower ramification groups>:
$$
\boxed{G_i=\{\sigma\in G:v_L(\sigma(\pi_L)-\pi_L)\geq i+1\}.}
$$
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