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  "body": "\n## Summary\n\nLet $L/K$ be a Galois extension of number fields, and let $\\mathfrak{P}$ be a prime of $L$.\nThe decomposition group of $\\mathfrak{P}$ is the subgroup of the [[Galois group|0XTY86BZKY]] that maps $\\mathfrak{P}$ to itself.\nIt surjects onto the Galois group of the residue field extension, and the kernel of this map is the inertia group.\nThe decomposition group describes how $\\mathfrak{P}$ splits, and it is the home of the [[Frobenius element|H5145REFMW]].\nIt is the Galois group of the local extension obtained by completing at $\\mathfrak{P}$.\n\n## Explanations\n\n### L1 human\n\nThe Galois group of $L/K$ moves the prime ideals of $L$ that lie over a given prime of $K$ around among themselves.\nThe decomposition group of one of them is the set of symmetries that leave it in place.\nIf the prime splits into many pieces, few symmetries fix a given piece, and the decomposition group is small.\nIn $\\Q(i)/\\Q$, complex conjugation swaps the two primes $(2 + i)$ and $(2 - i)$ over $5$, so their decomposition groups are trivial.\nIt fixes the prime $(3)$, so the decomposition group of $(3)$ is the whole Galois group.\n",
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