Évariste Galois: genius, radical, duelist

The great mathematician Évariste Galois was the founder of group theory. He was born in the village of Bourg-la-Reine, just outside Paris, in 1811. He died in 1832 after a duel at the age of 20. His father was a cultivated man, a staunch Republican, who taught the boy a hatred of tyranny. Maire of the village under Louis XVIII, the older Galois consistently opposed the local priest.

Evariste’s early education was taken care of by his mother who taught him classics. He entered the Lycée Louis-le-Grand at the age of 12. The school still exists today, but at the time it was run by a teacher who was more a jailer than anything else. Most of the teachers were of the same disciplinarian disposition. They were not evil, only stupid; but stupidity was the thing that Galois was unable to forgive.

Galois’ mathematical awakening occurred around the age of fourteen or fifteen. He somehow managed to get hold of Legendre’s work on algebra, aimed mostly at professional mathematicians. He devoured it as if it were an interesting novel. In other studies, his results were mediocre or worse. His teachers decided to demote him, i.e. to make him repeat a class. He was forced to study rhetoric, Latin and Greek for another year. He was bored.

Around the age of 16 he had another disappointment. He failed the highly competitive exam to the École Polytechnique. One of the reasons for this may have been the way he did his mathematics: all in his head, finding it difficult to put numbers into words. His examiner was Cauchy, who was regarded by many at the time as the king of mathematicians. How could he have failed Galois? Maybe he had a bad day or, as Horace says, sometimes even Homer nods.

For Galois this was a disaster. Had he gained entry to the École Polytechnique, he would have been taken care of and his mathematics would have flourished. In fact he would have been part of a group that then had the highest concentration of mathematical talent in the whole world.

At the age of 17 he met at the Louis-le-Grand a teacher who completely devoted himself to the mathematical education of his pupils. He recognised Galois’ talent and even attempted to send his best pupil to the École Polytechnique without an entrance examination. Alas, he did not succeed. Not all teachers at Louis-le-Grand were supportive. Here are some of the comments he received:

“The mathematical madness dominates this boy. I think his parents had better let him take only mathematics. He is wasting his time here, and all he does is torment his teachers and get into trouble.”

All his short life Galois had to contend with misfortune. In July 1829 his father committed suicide, unable to cope with the local priest’s hate campaign against him. Some more teachers’ comments:

“This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research. He has communicated to me some results in applied analysis.” His marks for literature were less favourable.

“This is the only student who has answered me poorly; he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly.”

In 1829, Galois was accepted at the École Normale. In 1830 there was a riot outside the door, with barricades erected, but the head of the school simply locked the doors, not letting anybody out. Later that year the July Revolution put Louis Philippe on the throne. That did not in any way diminish the political conflicts that went on till the next major revolution in 1848.

In 1831 Poisson, another famous mathematician, asked Galois to submit his work on the theory of equations to the Academy of Sciences. Unfortunately, Poisson declared them incomprehensible.

In 1832, after repeatedly displaying his anti-royalist feelings, Galois was expelled from the school. Some time later he was arrested for proposing a toast to Louis-Philippe with a knife in his hand, which was interpreted as a threat to the king’s life. Soon afterwards he was arrested again for wearing a forbidden republican uniform. Very soon after his release he got into another controversy with a “patriot” who challenged him to a duel. Under unclear circumstances, he accepted the challenge. The duel was fought the following morning at dawn. Galois was shot in the intestines and soon died of his wounds.

He wrote his “mathematical testament” on the night before the duel. Sixty pages of this and other papers are all that survived him. Fourteen years later Liouville, another great mathematician, revisited those proofs, filled a few gaps here and there and made those theorems more widely known. If his contemporaries did not sufficiently appreciate him, posterity certainly did. An entire branch of abstract algebra, Galois Theory, is named after him. Today, Évariste Galois is still regarded as one of the greatest mathematicians of all time.

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