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  "body": "\n## Statement\n\nThe angle $\\pi/3$, that is, $60^\\circ$, cannot be [[trisected|00J3DCCEK5]] with straightedge and compass.\nEquivalently, $\\cos(\\pi/9)$ is not a [[constructible number|0FMN2MN7TM]].\n\n## Remarks\n\nThe proof: $\\cos 3\\theta = 4 \\cos^3 \\theta - 3 \\cos \\theta$ shows that $c = \\cos(\\pi/9)$ is a root of $8x^3 - 6x - 1$. Then $u = 2c$ is a root of $x^3 - 3x - 1$.\nThe polynomial $x^3 - 3x - 1$ has no rational root, by the [[rational root theorem|0819CQ1V8H]]. So it is [[irreducible|0VNE161QCK]], and $[\\mathbb{Q}(c) : \\mathbb{Q}] = 3$.\nSince $3$ is not a power of $2$, the number $c$ is not constructible.\nThe angle $\\pi/3$ itself is constructible, so no general trisection method exists.\n\n## History\n\nPierre Wantzel published the proof in 1837.\n\n## Explanations\n\n### L1 human\n\nIf you could trisect every angle, you could trisect $60^\\circ$ and construct an angle of $20^\\circ$.\nThe cosine of $20^\\circ$ has degree $3$ over the rationals, so it is not constructible.\n",
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