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  "body": "\n## Summary\n\nAn algebraic set is the set of common zeros, in the affine space $k^n$ over a field $k$, of a family of polynomials.\nThe polynomials are in $n$ variables.\nAlgebraic sets are the basic geometric objects of classical algebraic geometry.\nThey are the closed sets of the Zariski topology on $k^n$.\nOver an algebraically closed field they correspond to radical ideals of the polynomial ring.\n\n## Explanations\n\n### L1 human\n\nWrite down some polynomial equations in $x_1, \\dots, x_n$ and look at all points that solve every one of them at once.\nThat set of solutions is an algebraic set.\nIn the plane, the equation $x^2 + y^2 - 1 = 0$ gives the unit circle, and the two equations $y = 0$ and $x = 0$ together give just the origin.\nLines, conics, finite sets of points and the whole plane are algebraic sets, but a disc or a half-line is not.\n",
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