= Ramification group
{title2=$G_i$}
{wiki}
= Higher ramification group
{synonym}
For a finite <Galois extension> $L/K$ of non-Archimedean <local fields>, normalize $v_L$ to have value group $\mathbb Z$. For integers $i\geq0$, the lower <ramification groups> are
$$
G_i=\{\sigma\in\operatorname{Gal}(L/K):v_L(\sigma(a)-a)\geq i+1\text{ for all }a\in\mathcal O_L\}.
$$
The group $G_0$ is the <inertia group>, whose automorphisms fix the residue field. The group $G_1$ is the <wild inertia group>, a group of prime-power order for the residue characteristic. Larger indices demand closer agreement with the identity on integral elements.
An <unramified extension> has $G_0=1$. A nontrivial totally and <tamely ramified extension> has $G_0=\operatorname{Gal}(L/K)$ and $G_1=1$. In a wild extension further groups can survive, and their sizes compute the <different exponent from ramification groups>. The <upper ramification numbering> reindexes this filtration to make it compatible with quotient Galois groups.
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