= Abel-Ruffini theorem
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The <Abel-Ruffini theorem> states that there is no formula using finitely many field operations and extraction of radicals that solves the general <quintic equation>. The same impossibility holds for the general <polynomial equation> of any <degree of a polynomial> at least five. It does not say that every individual <quintic equation> is insoluble by radicals.
In <Galois theory>, a characteristic-zero <polynomial> is solvable by radicals exactly when its <Galois group> is a <solvable group>. The general degree-$n$ <polynomial> has <symmetric group> $S_n$, which is not solvable for $n\ge5$. In contrast, $x^5-2=0$ has roots $\sqrt[5]2\,\zeta_5^j$, so it is solvable by radicals. <Niels Henrik Abel>'s 1824 memoir established the general quintic impossibility; its title states precisely that goal.<Mémoire sur les équations algébriques by Niels Henrik Abel>[1824 pamphlet, title and principal impossibility statement]{ref}
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