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  "body": "\n## Definition\n\nLet $L / K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of non-Archimedean [[local fields|J85FZEQG38]], with [[Galois group|0XTY86BZKY]] $G$.\nLet $\\mathcal{O}_L$ be the valuation ring of $L$ and let $v_L$ be the normalized discrete valuation of $L$, with $v_L(L^\\times) = \\Z$.\nFor an integer $i \\ge -1$, the $i$-th *ramification group* (in the lower numbering) is\n$$G_i = \\set{\\sigma \\in G : v_L(\\sigma(a) - a) \\ge i + 1 \\text{ for all } a \\in \\mathcal{O}_L}.$$\n\n## Remarks\n\nEach $G_i$ is a normal [[subgroup|0X1JXT47GA]] of $G$, and $G_{-1} = G \\supseteq G_0 \\supseteq G_1 \\supseteq \\cdots$.\nThe group $G_0$ is the inertia group, and $G_i = 1$ for $i$ large.\nIf $p$ is the characteristic of the [[residue field|NJ3APQ8BHH]], then $G_0 / G_1$ is cyclic of order prime to $p$, and $G_i / G_{i+1}$ is an elementary abelian $p$-group for $i \\ge 1$.\nIf $\\mathcal{O}_L = \\mathcal{O}_K[\\alpha]$, then $\\sigma \\in G_i$ exactly when $v_L(\\sigma(\\alpha) - \\alpha) \\ge i + 1$.\nExample: for $L = \\Q_2(i)$ and $K = \\Q_2$, one has $\\mathcal{O}_L = \\Z_2[i]$ and $v_L(2) = 2$.\nFor complex conjugation $\\sigma$, $v_L(\\sigma(i) - i) = v_L(-2i) = 2$, so $\\sigma \\in G_1$ and $\\sigma \\notin G_2$.\nNon-example: if $L / K$ is unramified and $\\sigma \\neq 1$, then $\\sigma$ acts nontrivially on the residue field, so $v_L(\\sigma(a) - a) = 0$ for some $a$ and $\\sigma \\notin G_0$.\nThere is also an upper numbering $G^u$, defined through the Herbrand function, which behaves well under quotients.\n",
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