This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.
For centuries the question was practical: given an equation, what formula, built from its coefficients using addition, multiplication and roots, gives its solutions? Formulas existed for equations up to the fourth power; a fifth-power equation resisted every attempt, and Niels Henrik Abel had shown by 1824 that no general formula could exist for it. What Abel had not shown was why, for some particular fifth-power equations, a formula does exist while for others it does not. Evariste Galois answered that, in his late teens, by asking a question nobody had thought to ask in quite that way: instead of studying the equation directly, study the ways its roots can be permuted among themselves while preserving every algebraic relationship between them. That collection of permutations has its own structure, closed under combination, with an identity and inverses, the object now called a group; and Galois showed that whether the equation is solvable by radicals is written entirely in the structure of that group, not in the equation's coefficients directly. Arthur Cayley would give the definition its modern abstract form in 1854, stripped of any particular equation, applicable to symmetry of every kind mathematics studies. But the idea that symmetry itself could be an object of study, with its own internal architecture worth analyzing on its own terms, is Galois', discovered to answer one narrow question about polynomials and found, ever since, to answer an enormous number of others.