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A 350-year-old problem, one night
Along the way he threatened the King of France twice and joined a paramilitary movement bent on toppling the government. The duel that ended his life was fought for the sake of a woman.
History's knots were tied by people in their 20s and early 30s doing their Great Work while their spirit burned hot. Galois did his before he turned 21.
How did a single letter save his work?
On the night of May 29, 1832, Galois wrote a letter to his friend Auguste Chevalier summarizing his discoveries and asking that they be published. Chevalier saw that it appeared in the Revue Encyclopédique later that year. The papers then sat largely unread until Joseph Liouville edited and published them in 1846. Camille Jordan's 1870 treatise on substitutions and equations then turned Galois's ideas into a systematic theory.
Why did group theory solve an equation problem?
Galois attached to every polynomial equation a group made of the permutations of its roots that preserve the algebraic relations between them. An equation can be solved using only arithmetic and roots if that group has a particular layered structure. The general fifth-degree equation has a group without that structure, which explains why Niels Henrik Abel's 1824 proof found no general formula for it. Groups later became a basic tool in physics and chemistry.
Why did the Academy overlook his early papers?
A manuscript Galois submitted to the French Academy of Sciences in 1829 was never presented, and the copy he sent for a mathematics prize in 1830 disappeared after the secretary, Joseph Fourier, died. In 1831 Siméon Denis Poisson reviewed a third version and found the argument not sufficiently clear or developed. The Academy rejected it on that report.
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