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  "body": "\n## Definition\n\nLet $(R, \\mathfrak m)$ be a [[local ring|MZ02BKPHJA]].\nThe *residue field* of $R$ is the [[quotient ring|0076HWHGHQ]] $k = R/\\mathfrak m$.\nIt is a [[field|04C5ZBG1WT]] because $\\mathfrak m$ is a [[maximal ideal|0PV74TR5FP]].\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and $\\mathfrak p$ a [[prime ideal|00MHHZ9SNE]].\nThe *residue field of $R$ at $\\mathfrak p$* is the residue field\n$$\\kappa(\\mathfrak p) = R_{\\mathfrak p} / \\mathfrak p R_{\\mathfrak p}$$\nof the [[localization|X6JD4AG973]] of $R$ at $\\mathfrak p$.\n\n## Remarks\n\nThe field $\\kappa(\\mathfrak p)$ is isomorphic to the [[field of fractions|0F2PTGX1A2]] of the [[integral domain|0QVD7WBFGN]] $R/\\mathfrak p$.\n\nExample: the residue field of $\\Z_{(p)}$ is $\\Z/p\\Z$, since $a/b \\mapsto a b^{-1} \\bmod p$ is a surjection with kernel $p\\Z_{(p)}$.\nThe residue field of $k[\\![x]\\!]$ is $k$, by taking the constant term.\nFor $R = \\Z$, the residue field at $(0)$ is $\\Q$ and the residue field at $p\\Z$ is $\\Z/p\\Z$.\n\nNon-example: in the local ring $\\Z_{(p)}$, the quotient $\\Z_{(p)}/p^2\\Z_{(p)} \\cong \\Z/p^2\\Z$ is not a field, because $p \\ne 0$ there and $p \\cdot p = 0$.\nOnly the quotient by the maximal ideal is a field.\n",
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