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  "body": "\n## Summary\n\nThe ideal of a subset $X$ of affine space $k^n$ is the set of all polynomials that vanish at every point of $X$.\nIt is the algebraic shadow of $X$ and is always a radical ideal.\nOver an algebraically closed field, Hilbert's Nullstellensatz says that this gives a bijection from algebraic sets to radical ideals.\nThis turns geometry into algebra.\n\n## Explanations\n\n### L1 human\n\nGiven a set of points, collect every polynomial that is zero on all of them.\nThese polynomials form an ideal: sums of such polynomials and multiples of them still vanish on the set.\nFor the $x$-axis in the plane, the ideal consists of all polynomials divisible by $y$.\nFor the single point $(1, 2)$, the ideal is generated by $x - 1$ and $y - 2$.\n",
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