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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]], let $n \\ge 1$, and let $I$ be an [[ideal|096Z9MP2ME]] of the [[polynomial ring|0SDC934R52]] $k[x_1, \\ldots, x_n]$.\nLet $V(I) = \\set{a \\in k^n : f(a) = 0 \\text{ for all } f \\in I}$ be the zero set of $I$.\nLet $I(V(I)) = \\set{f \\in k[x_1, \\ldots, x_n] : f(a) = 0 \\text{ for all } a \\in V(I)}$ be the ideal of polynomials vanishing on it.\nThen\n$$I(V(I)) = \\sqrt{I},$$\nwhere $\\sqrt{I} = \\set{f : f^m \\in I \\text{ for some } m \\ge 1}$ is the [[radical|437YRCNYRJ]] of $I$.\n\n## Remarks\n\nThe inclusion $\\sqrt{I} \\subseteq I(V(I))$ is clear: if $f^m$ vanishes at a point, so does $f$.\nFor the other inclusion, let $f$ vanish on $V(I)$ and add a new variable $y$ (the Rabinowitsch trick).\nThe ideal generated by $I$ and $1 - y f$ in $k[x_1, \\ldots, x_n, y]$ has no zeros, so it is the unit ideal by the [[weak Nullstellensatz|ZWS0YWGF17]].\nWriting $1$ as a combination and substituting $y = 1/f$, then clearing denominators, gives $f^m \\in I$ for some $m$.\n\nExample: in $\\C[x]$, let $I = (x^2)$; then $V(I) = \\set{0}$ and $I(V(I)) = (x) = \\sqrt{(x^2)}$.\n\nThe field must be algebraically closed: over $\\R$, let $I = (x^2 + 1)$ in $\\R[x]$.\nThen $V(I)$ is empty, so $I(V(I)) = \\R[x]$, but $\\sqrt{I} = I$ is a proper ideal, since $I$ is prime.\n\n## Explanations\n\n### L1 human\n\nA polynomial vanishes at all common zeros of some polynomials exactly when one of its powers is a combination of them.\nSo, over $\\C$, systems of polynomial equations and ideals of polynomials carry the same information, up to taking radicals.\n\n### L2 human\n\nThe theorem gives an inclusion-reversing bijection between radical ideals of $k[x_1, \\ldots, x_n]$ and algebraic subsets of $k^n$.\nPrime ideals correspond to irreducible algebraic sets and maximal ideals to points.\nIt is the bridge between commutative algebra and classical algebraic geometry.\nIn ring-theoretic form it says that $k[x_1, \\ldots, x_n]$ is a [[Jacobson ring|7EB64ZBXAT]]: every radical ideal is an intersection of maximal ideals.\n",
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