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  "body": "\n## Statement\n\nLet $q$ be an odd prime power, let $\\F_q$ be the [[finite field|0ZXZ82FTX2]] with $q$ elements, and let $\\mathrm{PG}(2, q)$ be the [[projective plane over $\\F_q$|CSMB49BDPT]].\nAn [[oval|7B03TB64N9]] of $\\mathrm{PG}(2, q)$ is a set of $q + 1$ points with no three on a common line.\nThen every oval of $\\mathrm{PG}(2, q)$ is a non-degenerate [[conic|R3TMVHTP3A]].\n\n## Remarks\n\nThe theorem was proved by Beniamino Segre.\nThe key step is the \"lemma of tangents\" for an oval in $\\mathrm{PG}(2, q)$ with $q$ odd.\nIt says that any triangle of points of the oval and the triangle of tangent lines at these points are in perspective.\nIts proof uses that the product of all nonzero elements of $\\F_q$ is $-1$.\nFrom this one shows that the oval agrees with the conic through five of its points.\n\nThe hypothesis that $q$ is odd is necessary.\nLet $q = 2^h$ with $h \\ge 3$, let $C$ be a conic and let $N$ be its nucleus, the common point of all its tangent lines.\nRemoving a point $P \\ne N$ of $C$ from $C \\cup \\set{N}$ gives an oval.\nIt is not a conic: it shares $q \\ge 8$ points with $C$, while two distinct conics share at most $4$ points.\n\n## Explanations\n\n### L1 human\n\nIn a finite plane over a field with an odd number of elements, take $q + 1$ points with no three on a line.\nThis is the largest such set possible, and conics are one way to get it.\nSegre's theorem says that conics are the only way.\nSo a purely combinatorial condition forces an algebraic shape, a curve of degree two.\n\n### L2 human\n\nSegre's theorem is the starting point of Galois geometry, the study of point sets in projective spaces over finite fields.\nIt shows that in $\\mathrm{PG}(2, q)$ with $q$ odd the ovals have a single type up to projective equivalence.\nIn even characteristic the picture is different: every oval extends to a hyperoval of $q + 2$ points, and many non-conic examples exist.\nThe same circle of ideas classifies ovoids of $\\mathrm{PG}(3, q)$ for $q$ odd as elliptic quadrics.\n",
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