Evariste Galois

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.

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

Evariste Galois died at twenty, on 31 May 1832, of a wound from a duel fought the previous morning; historians still disagree about who his opponent was and what the quarrel was actually about, with theories running from a political conflict of the turbulent post-1830 republican underground he had been jailed for supporting, to a private affair of the heart. The romantic version of the story, repeated for generations, has him staying up the whole night before the duel frantically writing down his mathematical results in case he did not survive, annotating a manuscript with the margin note "I have not time." Later historians of mathematics have found that account overstated: Galois had already submitted much of the relevant work to the Academie des Sciences years earlier, only to have it lost or rejected by referees including Cauchy and Poisson, so the final night was less a first draft of genius under pressure than a last attempt to make sure ideas already refused a hearing would not vanish with him. Either way, the manuscripts survived, in the hands of his brother Alfred, and Joseph Liouville edited and published them fourteen years later, in 1846, giving the mathematical world access to a theory that a young man's death had nearly cost it entirely.