Galois Wrote Modern Algebra the Night Before His Duel

On the morning of May 30, 1832, Évariste Galois walked into a field outside Paris and was shot in the abdomen. His opponent and seconds left him there. A passing farmer found him, and he died the next morning at the Hôpital Cochin, age twenty. His last words, recorded by his younger brother Alfred, were "Ne pleure pas, Alfred. J'ai besoin de tout mon courage pour mourir à vingt ans." Don't cry — I need all my courage to die at twenty.

The duel's cause is still murky. Five days before, Galois had written to his friend Auguste Chevalier alluding to a broken love affair. Some accounts blame political enemies; he had spent months in prison for republican agitation. The historical record won't quite resolve it.

What is settled is what he did the night before. Knowing he might die, Galois sat down and wrote a long letter to Chevalier summarizing his mathematics. He included theorems with sketches of proofs, conjectures he wouldn't get to verify, and instructions for what to do with his manuscripts. "Tu prieras publiquement Jacobi ou Gauss," he wrote — ask Jacobi or Gauss publicly to give an opinion. The letter runs to about a dozen pages.

The core idea was this. For more than three hundred years, mathematicians had a formula for solving quadratics, then cubics (Cardano, 1545), then quartics (Ferrari, around the same time). Each formula gives the roots in terms of the coefficients using nothing but addition, subtraction, multiplication, division, and roots — what's called "solving by radicals." Then the trail went cold. No one could find a quintic formula.

Galois did not find one. He proved one cannot exist for the general quintic, and went further: he gave a method to decide, for any polynomial, whether its roots can be written using radicals. The method depended on attaching to each polynomial a finite group — what we now call its Galois group — built from the ways the roots can be permuted while preserving algebraic relations. The polynomial is solvable by radicals if and only if that group is "solvable" in a precise technical sense involving its tower of normal subgroups.

Niels Henrik Abel, working in Norway, had proved the quintic case in 1824 by a different route, and died of tuberculosis in 1829 before learning Galois existed. What Galois added was the general framework: not just "the quintic isn't solvable," but a structural theory that explained why some equations are and others aren't. Group theory, today the language of symmetry across mathematics and physics, is what fell out.

The École Polytechnique had rejected Galois twice. He had submitted a paper on the theory to the Paris Academy in 1829; the referee, Cauchy, lost or set aside the manuscript. He submitted a revised version for the Academy's Grand Prize in 1830; the referee was Fourier, who took the paper home and died before reading it. A third submission was returned by Poisson with a note that the proofs were unclear. By the duel, none of his work was in print.

Chevalier kept the letter. In 1843, eleven years after Galois died, Joseph Liouville read through the manuscripts and recognized what was there. He published an edited version in 1846 in the Journal de Mathématiques Pures et Appliquées. By the 1870s, Camille Jordan's Traité had absorbed and extended Galois's ideas into a full theory, and group theory was on its way to becoming foundational.

Hermann Weyl later wrote that the Chevalier letter, judged by the novelty and depth of the ideas it contains, "is perhaps the most substantial piece of writing in the whole literature of mankind." That sounds like hyperbole until you remember that he had eight hours and was bleeding to death the next day.

The story is sometimes told as if Galois invented group theory in a night, scribbling "je n'ai pas le temps" — I don't have time — in the margins. Both details are real but slightly misleading. He had been working on the theory for years; the letter recapped and clarified, it didn't conjure. And the marginal note appears in a draft from earlier in the year. What he did finish that night was the explicit framing — the part that future readers, including Liouville, could follow. Without that framing, the manuscripts in Chevalier's drawer might have stayed unreadable. Without the duel, he might have written a textbook by thirty.