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  "body": "\n## Statement\n\nLet $L/K$ be a finite extension of [[algebraic number fields|MWBDFS2KJ5]] of [[degree|0EAJ2XXADC]] $n$, with rings of integers $\\mathcal{O}_K \\subseteq \\mathcal{O}_L$.\nLet $\\mathfrak{p}$ be a nonzero prime ideal of $\\mathcal{O}_K$, with [[factorisation|B7KNZCBFYT]] $\\mathfrak{p}\\mathcal{O}_L = \\mathfrak{P}_1^{e_1} \\cdots \\mathfrak{P}_g^{e_g}$ into distinct prime ideals.\nLet $f_i = [\\mathcal{O}_L/\\mathfrak{P}_i : \\mathcal{O}_K/\\mathfrak{p}]$ be the residue degrees.\nThen\n$$\\sum_{i=1}^{g} e_i f_i = n.$$\nIf moreover $L/K$ is a [[Galois extension|0QMZAQ6QQ4]], then $e_1 = \\cdots = e_g = e$ and $f_1 = \\cdots = f_g = f$, and $efg = n$.\n\n## Remarks\n\nProof idea: count elements of $\\mathcal{O}_L/\\mathfrak{p}\\mathcal{O}_L$.\nIt is a vector space over $\\mathcal{O}_K/\\mathfrak{p}$ of dimension $n$, by localising at $\\mathfrak{p}$, where $\\mathcal{O}_L$ becomes free of rank $n$.\nBy the Chinese remainder theorem it is the product of the rings $\\mathcal{O}_L/\\mathfrak{P}_i^{e_i}$, and each has dimension $e_i f_i$, using the filtration by powers of $\\mathfrak{P}_i$.\nIn the Galois case, the Galois group acts transitively on the $\\mathfrak{P}_i$, and conjugate primes have the same $e$ and $f$.\n\nExample: in $\\Q(\\sqrt{-5})/\\Q$, with $n = 2$: the prime $2$ has $e = 2, f = 1, g = 1$; the prime $3$ has $e = 1, f = 1, g = 2$; the prime $11$ is inert since $-5$ is not a square modulo $11$, with $e = 1, f = 2, g = 1$.\nIn each case $efg = 2$.\n\n## Explanations\n\n### L1 human\n\nWhen a prime ideal of a small field breaks into primes of a larger field, the pieces must add up to the degree of the extension.\nEach piece counts with its exponent times the size of its residue field extension.\nSo a prime in a quadratic field either splits into two, stays prime with a bigger residue field, or becomes a square.\n\n### L2 human\n\nThe identity limits the possible splitting types and is used constantly in computations.\nIn a Galois extension, $e f$ is the order of the [[decomposition group|7X92G8CQ33]] of any prime over $\\mathfrak{p}$, and $e$ is the order of its inertia group.\nFor a cyclic extension of prime degree $\\ell$, the identity leaves only three cases: split completely ($g = \\ell$), inert ($f = \\ell$) or totally ramified ($e = \\ell$).\n",
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