Galois linked equation-solving to the symmetry of roots
Évariste Galois showed that whether a polynomial can be solved by radicals depends on the structure of its root-permutation group. That bridge between fields and groups also explains why some ancient constructions with compass and straightedge are impossible.
The fundamental theorem of Galois theory turns certain field-extension questions into group questions that are easier to handle. An equation is solvable by radicals when its roots can be written with integers, nth roots, and the four arithmetic operations. Galois characterized that property by a condition on the permutation group of the roots—today, whether the Galois group is a solvable group. The Abel–Ruffini theorem is the special case that a general degree-five-or-higher polynomial has no such formula; Galois explains which particular higher-degree equations still yield.
The same machinery settles classical geometry: doubling the cube and trisecting arbitrary angles are impossible as stated, and it completes Gauss's list of constructible regular polygons with a proof of completeness. Background runs through symmetric functions and Viète's formulas, Cardano and Ferrari's cubic and quartic solutions in the 1545 Ars Magna, and Lagrange's 1770 analysis of those solutions via root permutations—without composing permutations. Ruffini (1799) and Abel (1824) blocked the general quintic; Galois, at eighteen in 1830, submitted the precise group-theoretic criterion. The Academy rejected the memoir as too sketchy in 1831; Galois died in a duel in 1832; Liouville published the paper in 1846 after announcing the result in 1843.
Contemporaries found the theory hard. Liouville's commentary missed the group core; Serret's 1866 textbook and Jordan's 1870 Traité spread understanding in France; Dedekind lectured on it at Göttingen in 1858; Netto and Weber carried it to wider German and American audiences. Later generalizations include Galois connections and Grothendieck's Galois theory. The teenage insight still decides, algorithmically, which equations surrender to radicals.
Source: Galois theory
