Insolvability of the quintic

There is no general formula — built from the coefficients of a degree-5 (or higher) polynomial equation using addition, subtraction, multiplication, division, and $n$-th roots (radicals) alone — that expresses its roots, unlike the formulas that exist for degree 2, 3, and 4.

Paolo Ruffini published an attempted proof in 1799 (expanded 1813) that no radical formula solves the general quintic, but his argument had significant gaps. The first complete, rigorous proof — the Abel–Ruffini theorem — is due to Niels Henrik Abel, who published it at his own expense in 1824 and gave a fuller version in Crelle's Journal in 1826. A few years later, Évariste Galois went further: he attached to every polynomial equation a group of permutations of its roots (its Galois group) and proved that the equation is solvable by radicals exactly when that group is solvable. This explains not only why no single formula works for every degree-5 equation, but exactly which individual equations, of any degree, can and cannot be solved by radicals.