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  "body": "\n## Definition\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|00MHHZ9SNE]] of $\\mathcal{O}_K$.\nSince $\\mathcal{O}_L$ is a [[Dedekind domain|7NPY2CN5KB]], the ideal $\\mathfrak{p}\\mathcal{O}_L$ factors uniquely as\n$$\\mathfrak{p}\\mathcal{O}_L = \\mathfrak{P}_1^{e_1} \\cdots \\mathfrak{P}_g^{e_g}$$\nwith distinct nonzero prime ideals $\\mathfrak{P}_i$ of $\\mathcal{O}_L$ and integers $e_i \\ge 1$.\nThe $\\mathfrak{P}_i$ are the primes of $L$ *lying over* $\\mathfrak{p}$.\nThe number $e_i$ is the [[ramification index|H5RC67SD24]] of $\\mathfrak{P}_i$ over $\\mathfrak{p}$, and the *residue degree* (or inertia degree) is $f_i = [\\mathcal{O}_L/\\mathfrak{P}_i : \\mathcal{O}_K/\\mathfrak{p}]$.\nThe prime $\\mathfrak{p}$ is said to:\n\n1. *split completely* in $L$ if $g = n$, that is, $e_i = f_i = 1$ for all $i$;\n2. be *inert* in $L$ if $\\mathfrak{p}\\mathcal{O}_L$ is a prime ideal, that is, $g = 1$ and $e_1 = 1$;\n3. be *ramified* in $L$ if $e_i \u003e 1$ for some $i$.\n\n## Remarks\n\nThe integers satisfy $\\sum_{i=1}^g e_i f_i = n$, the [[fundamental identity|REJDQ6BTQV]].\nIf $L/K$ is a [[Galois extension|0QMZAQ6QQ4]], the Galois group permutes the $\\mathfrak{P}_i$ transitively, so all $e_i$ are equal to some $e$, all $f_i$ to some $f$, and $efg = n$.\n\nExample: for $K = \\Q$, $L = \\Q(i)$ and $\\mathfrak{p} = 5\\Z$, one has $5\\Z[i] = (2+i)(2-i)$ with two distinct primes, each with residue field $\\F_5$.\nSo $g = 2 = n$ and $5$ splits completely.\n\nNon-example: $3$ does not split completely in $\\Q(i)$.\nThe ideal $3\\Z[i]$ is prime, because $x^2 + 1$ is irreducible modulo $3$; so $g = 1$, $f = 2$, and $3$ is inert.\nThe prime $2$ is ramified: $2\\Z[i] = (1+i)^2$, with $e = 2$.\n",
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