Galois Theory

Galois theory is the branch of abstract algebra that establishes a correspondence between the solvability of polynomial equations and the structure of permutation groups. Developed by Évariste Galois in the early 1830s, it is the canonical example of a mathematical problem being solved not by manipulating the objects in question but by translating them into a structural framework where the answer becomes visible.

The central theorem: a polynomial equation is solvable by radicals — by nested root extractions — if and only if its Galois group has a composition series with cyclic quotients. The quintic equation cannot be solved by radicals because its Galois group, the symmetric group S₅, is not solvable in this sense. The equation itself is unchanged; what changed was the theory available to interpret it.

Galois theory marks a turning point in the history of mathematics: the moment when algebra ceased to be the study of numbers and became the study of structure. It is the ancestor of group theory, field theory, and much of modern algebraic geometry.