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  "body": "\n## Definition\n\nLet $X$ be a [[scheme|5A35TYZD02]] and let $x \\in X$.\nThe [[stalk|JTAKAPTT8H]] $\\mathcal{O}_{X,x}$ is a [[local ring|MZ02BKPHJA]]; let $\\mathfrak{m}_x$ be its maximal ideal.\nThe *residue field* of $X$ at $x$ is the field\n$$\\kappa(x) = \\mathcal{O}_{X,x} / \\mathfrak{m}_x.$$\nFor an open set $U \\ni x$ and $s \\in \\mathcal{O}_X(U)$, the *value* $s(x)$ is the image of $s$ in $\\kappa(x)$.\n\n## Remarks\n\nExample: for $X = \\operatorname{Spec} A$ and a prime ideal $\\mathfrak{p}$ of $A$, the residue field is $\\kappa(\\mathfrak{p}) = A_{\\mathfrak p} / \\mathfrak{p} A_{\\mathfrak p}$.\nIt is isomorphic to the field of fractions of $A / \\mathfrak{p}$.\nFor $A = \\Z$ this gives $\\kappa((p)) = \\F_p$ for a prime $p$ and $\\kappa((0)) = \\Q$.\nFor $A = \\R[x]$ and $\\mathfrak{p} = (x^2 + 1)$ it gives $\\R[x]/(x^2+1) \\cong \\C$, so this point has residue field larger than $\\R$.\nNon-example: the stalk $\\mathcal{O}_{X,x}$ itself is in general not a field; for $\\Z$ at $(p)$ it is $\\Z_{(p)}$, which has the non-unit $p \\ne 0$.\nFor a field $K$, morphisms $\\operatorname{Spec} K \\to X$ correspond to pairs of a point $x \\in X$ and a field embedding $\\kappa(x) \\to K$.\n",
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