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  "body": "\n## Example\n\nA quintic equation is an equation $ax^5 + bx^4 + cx^3 + dx^2 + ex + g = 0$ with $a \\ne 0$.\nConsider the quintic $f(x) = x^5 - 4x + 2$ over $\\Q$.\n\n1. $f$ is irreducible over $\\Q$ by [[Eisenstein's criterion|02BPKD7TMD]] at $2$, as $2$ divides $-4$ and $2$ but not $1$, and $4$ does not divide $2$.\n2. $f(-2) = -22$, $f(0) = 2$, $f(1) = -1$ and $f(2) = 26$, so by the [[intermediate value theorem|00KAFBR32E]] $f$ has at least three real roots.\n3. $f'(x) = 5x^4 - 4$ has exactly two real zeros, so by [[Rolle's theorem|0194DY35BG]] $f$ has at most three real roots.\n4. Hence $f$ has exactly three real roots and two nonreal roots.\n\nBy the [[theorem on prime degree polynomials with two nonreal roots|4CMPPTSJNP]], the Galois group of $f$ over $\\Q$ is $S_5$.\nSince [[$S_5$ is not solvable|0J53NE674D]], [[Galois' theorem on solvability|VHCWCFA3MX]] shows that $f$ is not [[solvable by radicals|J7YS0DA3HQ]].\nBy contrast, $x^5 - 2$ is solvable by radicals: its roots are $\\sqrt[5]{2}\\, \\zeta^k$, with $\\zeta = e^{2\\pi i/5}$ and $k = 0, \\dots, 4$.\n\n## Explanations\n\n### L1 human\n\nThe equation $x^5 - 4x + 2 = 0$ has five complex solutions, and they can be computed to any accuracy.\nBut none of them can be written using only whole numbers, arithmetic and $n$-th roots.\nThis concrete equation shows that the [[Abel–Ruffini theorem|163A2WQQJX]] is about specific equations, not only about a general formula.\n",
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