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  "body": "\n## Statement\n\nLet $K$ be a [[subfield|07QY5RPFBR]] of $\\R$, and let $f \\in K[x]$ be a cubic polynomial that is [[irreducible|00X2A9G9W3]] over $K$ and has three real roots.\nThen no root of $f$ lies in a [[radical extension|V74WMGQQ9D]] $R$ of $K$ with $R \\subseteq \\R$.\n\n## Remarks\n\nThe three roots are distinct, since an irreducible polynomial over a field of characteristic $0$ has no repeated roots.\nFor the idea of the proof, adjoin first the square root of the discriminant, which is real.\nThen over every real field $F$ that contains this square root and over which $f$ is irreducible, adjoining one root of $f$ adjoins all three roots.\nTake a real tower of steps adjoining $q$-th roots, $q$ prime, whose last step $E \\subseteq E(\\alpha)$ with $\\alpha^q \\in E$ first makes $f$ reducible.\nThen $3$ divides $[E(\\alpha) : E]$, which is $1$ or $q$, so $q = 3$ and $E(\\alpha)/E$ is normal of degree $3$.\nHence $x^3 - \\alpha^3$ splits in $E(\\alpha)$, so this real field contains a nonreal cube root of unity, which is impossible.\nExample: $f = x^3 - 3x + 1$ over $\\Q$ is irreducible, since $\\pm 1$ are not roots, and its roots $2\\cos(2\\pi/9)$, $2\\cos(4\\pi/9)$, $2\\cos(8\\pi/9)$ are real.\nSo every expression of these roots by radicals passes through nonreal numbers, as in the [[cubic formula|FB6J5WAT5V]].\nIrreducibility is necessary: $x^3 - x = x(x - 1)(x + 1)$ has three real roots, all in $\\Q$.\n\n## Explanations\n\n### L1 human\n\nThe cubic formula sometimes takes the square root of a negative number even when all three answers are real.\nThe casus irreducibilis says this detour is unavoidable when the cubic does not factor.\nNo formula built only from real roots of real numbers can give the roots.\n\n### L2 human\n\nFor a real cubic with three distinct real roots, the discriminant is positive, so the square root in the [[cubic formula|FB6J5WAT5V]] is of a negative number.\nThe theorem shows that the problem is not the formula but the numbers themselves.\nThe roots can still be written with real numbers using trigonometry, for example as $2\\sqrt{-p/3}\\cos\\theta$ for suitable angles $\\theta$ when the cubic is $t^3 + pt + q$.\n",
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