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  "body": "\n## Definition\n\nLet $L/K$ be a finite [[Galois extension|0QMZAQ6QQ4]] of [[algebraic number fields|MWBDFS2KJ5]] with [[Galois group|0XTY86BZKY]] $G$, and let $\\mathcal{O}_K \\subseteq \\mathcal{O}_L$ be the rings of integers.\nLet $\\mathfrak{P}$ be a nonzero [[prime ideal|00MHHZ9SNE]] of $\\mathcal{O}_L$.\nThe *decomposition group* of $\\mathfrak{P}$ is the [[subgroup|0X1JXT47GA]]\n$$D_{\\mathfrak{P}} = \\set{\\sigma \\in G : \\sigma(\\mathfrak{P}) = \\mathfrak{P}}.$$\nThe *inertia group* of $\\mathfrak{P}$ is\n$$I_{\\mathfrak{P}} = \\set{\\sigma \\in D_{\\mathfrak{P}} : \\sigma(x) \\equiv x \\pmod{\\mathfrak{P}} \\text{ for all } x \\in \\mathcal{O}_L}.$$\n\n## Remarks\n\nLet $\\mathfrak{p} = \\mathfrak{P} \\cap \\mathcal{O}_K$, with ramification index $e$, residue degree $f$ and $g$ primes over it, as in [[splitting of prime ideals|B7KNZCBFYT]].\nThen $D_{\\mathfrak{P}}$ has order $ef$ and index $g$ in $G$.\nEach $\\sigma \\in D_{\\mathfrak{P}}$ induces an automorphism of the [[finite field|0ZXZ82FTX2]] $\\mathcal{O}_L/\\mathfrak{P}$ over $\\mathcal{O}_K/\\mathfrak{p}$; this gives a surjective homomorphism onto the Galois group of the residue extension, with kernel $I_{\\mathfrak{P}}$, so $\\abs{I_{\\mathfrak{P}}} = e$.\nFor $\\tau \\in G$, one has $D_{\\tau(\\mathfrak{P})} = \\tau D_{\\mathfrak{P}} \\tau^{-1}$.\n\nExample: in $\\Q(i)/\\Q$, with $G = \\set{1, c}$ where $c$ is complex conjugation, take $\\mathfrak{P} = (3)$.\nThen $c(\\mathfrak{P}) = \\mathfrak{P}$, so $D_{\\mathfrak{P}} = G$; and $c(i) = -i \\not\\equiv i \\pmod 3$, so $I_{\\mathfrak{P}}$ is trivial, matching $e = 1$, $f = 2$.\n\nNon-example: for $\\mathfrak{P} = (2 + i)$, the element $c$ is not in $D_{\\mathfrak{P}}$, because $c(\\mathfrak{P}) = (2 - i) \\ne \\mathfrak{P}$ (the quotient $(2+i)/(2-i)$ is not in $\\Z[i]$).\nSo $D_{\\mathfrak{P}}$ is trivial, matching $e = f = 1$ and $g = 2$.\n",
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