Abel-Ruffini theorem (1824): The general degree-5 polynomial has no solution by radicals. The proof, completed by Galois, shows the Galois group of a generic quintic is S₅, which is not solvable (its derived series S₅ ⊃ A₅ ⊃ {e} does not reach the trivial group — A₅ is the unique simple non-abelian group of order 60). Radical expressions correspond to cyclic group extensions, requiring a solvable tower. Polynomials with cyclic Galois group (e.g. xⁿ−1) are solvable. Shown: roots animate as coefficients vary; colors encode conjugacy classes.