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  "body": "\n## Definition\n\nLet $L/K$ be a finite extension of [[algebraic number fields|MWBDFS2KJ5]], with rings of integers $\\mathcal{O}_K \\subseteq \\mathcal{O}_L$.\nLet $\\mathfrak{p}$ be a nonzero [[prime ideal|00MHHZ9SNE]] of $\\mathcal{O}_K$, and let $\\mathfrak{P}$ be a prime ideal of $\\mathcal{O}_L$ with $\\mathfrak{P} \\cap \\mathcal{O}_K = \\mathfrak{p}$.\nThe *ramification index* $e(\\mathfrak{P} \\mid \\mathfrak{p})$ is the exponent of $\\mathfrak{P}$ in the factorisation of $\\mathfrak{p}\\mathcal{O}_L$ into prime ideals of the [[Dedekind domain|7NPY2CN5KB]] $\\mathcal{O}_L$.\nThe prime $\\mathfrak{P}$ is *ramified* over $\\mathfrak{p}$ if $e(\\mathfrak{P} \\mid \\mathfrak{p}) \u003e 1$.\nThe prime $\\mathfrak{p}$ is *ramified in $L$* if some prime of $\\mathcal{O}_L$ over $\\mathfrak{p}$ is ramified, and *unramified in $L$* otherwise.\n\n## Remarks\n\nFor number fields the residue field extensions are finite extensions of finite fields, hence separable, so no separability condition is needed.\nA prime number $p$ ramifies in a number field $L$ if and only if $p$ divides the [[discriminant|E0VZTNPJXA]] $d_L$; in particular only finitely many primes ramify.\nSee [[splitting of prime ideals|B7KNZCBFYT]] for the full description of the factorisation.\n\nExample: in $L = \\Q(\\sqrt{-5})$, the ideal $2\\mathcal{O}_L$ equals $\\mathfrak{P}^2$ with $\\mathfrak{P} = (2, 1 + \\sqrt{-5})$.\nSo $e(\\mathfrak{P} \\mid 2) = 2$ and $2$ is ramified, in line with $2 \\mid d_L = -20$.\n\nNon-example: $3$ is unramified in $\\Q(\\sqrt{-5})$.\nIndeed $x^2 + 5 \\equiv (x - 1)(x + 1) \\pmod 3$ has distinct roots, so $3\\mathcal{O}_L = (3, \\sqrt{-5} - 1)(3, \\sqrt{-5} + 1)$ with two distinct primes, each of exponent $1$; and $3 \\nmid 20$.\n",
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