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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 $n\ge 5$. 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.^[ref]

Niels Henrik Abel proved that the general quintic equation does not have solvability by radicals. His 1824 pamphlet established a limitation on formulas built from arithmetic operations and repeated extraction of roots, rather than a failure of numerical solution. Particular quintic equations can still have solvability by radicals; the impossibility concerns a formula for the general case.^[ref]

The result, expressed in the Abel-Ruffini theorem, helped shift the study of polynomial equations from searching for ever larger formulas toward asking which structural properties permit an algebraic solution.

A quintic equation is a one-variable polynomial equationOver the complex numbers, it has five roots of a polynomial counted with multiplicity of a root, by the fundamental theorem of algebra. Over the real numbers, at least one root is real because the leading odd-degree term makes the polynomial take opposite signs far enough in the two directions.

Some quintics are easy to solve: $x^5-x=x(x^2-1)(x^2+1)$ has roots $0,1,-1,i,-i$. The Abel-Ruffini theorem says that a formula by radicals cannot solve the general quintic equation; it does not rule out special factorizations, other special-function descriptions, or numerical root finding. Thus failure of a universal radical formula does not prevent exact or approximate solutions of particular equations.