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  "body": "\n## Summary\n\nThe residue field of a [[local ring|MZ02BKPHJA]] $R$ with maximal ideal $\\mathfrak m$ is the [[quotient ring|0076HWHGHQ]] $R/\\mathfrak m$, which is a [[field|04C5ZBG1WT]].\nMore generally, the residue field of any [[commutative ring|0310REJJY8]] at a [[prime ideal|00MHHZ9SNE]] $\\mathfrak p$ is the residue field of the [[localization|X6JD4AG973]] $R_{\\mathfrak p}$.\nIt is the field of \"values at a point\", and many invariants of local rings, such as the [[cotangent space|Z1CGB483J6]], are vector spaces over it.\n\n## Explanations\n\n### L1 human\n\nReducing modulo the maximal ideal throws away everything except the \"value at the point\".\nFor the ring $\\Z_{(p)}$ of fractions with denominators prime to $p$, reducing modulo $p$ gives the field $\\Z/p\\Z$ with $p$ elements.\nFor the power series ring $k[\\![x]\\!]$, it keeps only the constant term, so the residue field is $k$.\nThe [[definition|8ST7CTZ2DE]] gives the construction.\n",
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