Due to efforts of Abel, Ruffini, we know that there does not exist a general formula for a quintic equation. But given a specific quintic, say $x^5+5x^2-97x+1001=0$, Can there exist a root for it in radicals not necessarily involving the coefficients?
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Your example has Galois group $S_5$ (according to Maple), so its roots can't be expressed by radicals. – Robert Israel Feb 15 '24 at 19:03
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Radicals involving rational numbers, that is. – Robert Israel Feb 15 '24 at 19:09
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2But there are quintics that have roots expressed in radicals. For example, quintics that are not irreducible over the rationals. – Robert Israel Feb 15 '24 at 19:13