 Abel-Ruffini theorem

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]

The Abel–Ruffini theorem is a result in algebra that states there is no general solution in radicals to polynomial equations of degree five or higher. In other words, it is impossible to express the roots of a general polynomial of degree five or greater using only radicals (i.e., through a finite sequence of operations involving addition, subtraction, multiplication, division, and taking roots).