Évariste Galois (1811–1832) packed one of the deepest ideas in all of
mathematics into a life that lasted just twenty years. As a teenager he worked
out why some equations can be solved by a formula and others can never be — and in doing
so more or less invented
Galois was, by any ordinary measure, a disaster of a student. He failed the entrance exam to the École Polytechnique twice — the second time, legend has it, by throwing a blackboard eraser at an examiner who wouldn't accept his leaps of reasoning. Two separate manuscripts he sent to the Academy of Sciences were lost, and a third was rejected as too obscure. The examiners weren't entirely to blame: Galois saw so far ahead that he skipped the steps mere mortals needed.
Galois's leap was to stop staring at an equation's roots and start studying their
symmetries — the ways you can shuffle the roots around without breaking any true
relationship between them. Those shuffles form a group, and the shape of that
group decides everything. If the group can be taken apart in a certain tidy way (it is
In May 1832, aged 20, Galois was drawn into a duel — over a love affair, or a political trap, or both; the details are still argued. Convinced he would die, he spent the night before feverishly writing out his mathematics, scribbling "je n'ai pas le temps" — "I don't have the time" — in the margins. He was shot the next morning and died the day after. It took the world about forty years to catch up with what that 20-year-old had written in a single night.