= Solution
Normalize the <discrete valuation> on $L$ by $v_L(L^\times)=\mathbb Z$, set $v_L(0)=+\infty$, and write $\mathcal O_L$ for its <valuation ring>. For a finite <Galois extension> of <local fields>, define $G_{-1}=G=\operatorname{Gal}(L/K)$ and, for integers $i\geq0$,
$$
\boxed{G_i=\{\sigma\in G:v_L(\sigma(a)-a)\geq i+1\text{ for every }a\in\mathcal O_L\}.}
$$
Equivalently, $G_i$ is the kernel of the action on $\mathcal O_L/\mathfrak m_L^{i+1}$. These are the <lower ramification numbering> of the <ramification groups>. In particular, $G_0$ is the <inertia group>, the kernel of the action on the <residue field>, and $G_1$ is the <wild inertia group>. They form a decreasing sequence of <normal subgroups>, eventually trivial.
For a real index $t\geq-1$, extend by $G_t=G_{\lceil t\rceil}$. Define the <Herbrand function> by
$$
\varphi_{L/K}(t)=\int_0^t\frac{ds}{[G_0:G_s]}\quad(t\geq0),\qquad\varphi_{L/K}(t)=t\quad(-1\leq t\leq0).
$$
It is continuous, strictly increasing and piecewise linear; denote its inverse by $\psi_{L/K}$. The <upper ramification numbering> is
$$
\boxed{G^u=G_{\psi_{L/K}(u)}\qquad(u\geq-1).}
$$
The definition includes the placement of the groups at break endpoints: the group at a break is the group before the drop. Lower numbering is compatible with subgroups; upper numbering is the one compatible with quotients. In a <totally ramified extension>, the <uniformizer criterion for lower ramification groups> permits testing the defining <valuation> inequality on one <uniformizer> instead of on every integral element.
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