Mathematicians, Codes, and Obsessions That Approached the Human Limit

Évariste Galois: The Young Man Who Deciphered Equations Before Dying in a Duel

On the night before the duel that ended his life, Évariste Galois wrote a letter to his friend Auguste Chevalier summarizing several of his mathematical discoveries. He did not, however, create his entire theory that night, as the legend often claims. He had worked on it for years and used his final hours to organize, correct, and identify the ideas he feared would remain misunderstood.

He knew he would probably die at dawn, but he did not spend his final hours writing a sentimental farewell.

He wrote mathematics.

In the margins of some manuscripts, he left hurried remarks. On one of them, he wrote: “I have no time.” He was twenty years old and attempting to condense a new way of understanding equations before participating in a duel whose cause is still not known with certainty.

Évariste Galois died without knowing that those pages would help establish one of the central structures of modern mathematics.

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E. Galois Letter Évariste Galois 29 May 1832 This is the last page of letter from Évariste Galois, French mathematician, to his friend Auguste Chevalier.

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Wikimedia Commons

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Public domain

Iyanaga, Shokichi, "ガロアの時代 ガロアの数学 第一部 時代篇" , Springer-Verlag Tokyo, 1999

A Child Educated Outside School

Évariste Galois was born on October 25, 1811, in Bourg-la-Reine, near Paris (Toti Rigatelli, 1996; University of St Andrews, n.d.). His father, Nicolas-Gabriel Galois, was a liberal republican who became mayor of the town. His mother, Adélaïde-Marie Demante, had a strong education in classical literature and personally taught her son during his first twelve years.

Galois grew up studying Latin, Greek, philosophy, and rhetoric. Mathematics did not initially occupy an exceptional place in his education.

In 1823, he entered the Lycée Louis-le-Grand in Paris, a prestigious institution governed by strict discipline. There he encountered an educational system based heavily on memorization and obedience. His early results were uneven, and some teachers considered him distracted or undisciplined.

Everything changed when he began studying geometry.

Instead of relying solely on school textbooks, he read advanced works directly, particularly Adrien-Marie Legendre’s Elements of Geometry and Joseph-Louis Lagrange’s writings on equations. While his classmates learned procedures, Galois attempted to understand the entire architecture of a problem.

That way of thinking became both his greatest gift and a constant source of conflict.

The Examination He Could Not Explain

Galois wanted to enter the École Polytechnique, France’s most prestigious scientific and mathematical institution. He attempted its entrance examination in 1828 and failed (Rothman, 1982).

He tried again in 1829. He was rejected a second time.

Tradition transformed the second examination into an almost theatrical scene: an examiner unable to understand him, an absurd question, and Galois throwing an eraser at the professor. There is not enough documentation to accept all these details as historical fact.

What appears more likely is that Galois reasoned too rapidly and unconventionally for the oral format. He omitted steps he considered obvious and became impatient when asked to explain elementary procedures.

His failure does not prove that the examiners were incompetent or that Galois lacked mathematical knowledge. It reveals something more complicated: possessing extraordinary ideas does not guarantee an ability to communicate them within an institution.

He eventually entered the École Préparatoire, later incorporated into the École Normale. He considered it an inferior alternative, but continued developing his research there.

The Question That Had Resisted for Centuries

Since the Renaissance, mathematicians had known formulas for solving equations of the second, third, and fourth degrees using arithmetic operations and roots.

The great question was whether a similar formula could be found for every equation of the fifth degree.

Paolo Ruffini and Niels Henrik Abel had already demonstrated that no general formula by radicals existed for all quintic equations. Galois carried the problem much further (Stewart, 2015).

Instead of searching for another formula, he asked what internal structure determines whether a particular equation can be solved by radicals.

To answer that question, he studied the possible permutations of its roots. He discovered that these transformations could be organized into a system governed by precise rules (Galois, 2011; Stewart, 2015). He called these structures groups, using the word in a sense close to its present meaning in algebra.

The idea was revolutionary: understanding an equation did not always require directly calculating its solutions. One could instead study the network of symmetries connecting them.

Galois transformed a problem of calculation into a problem of structure.

Manuscripts That Could Not Find a Reader

Recognizing the importance of these ideas was not easy. Galois wrote in a compressed style, omitted intermediate demonstrations, and treated connections as obvious when other mathematicians could not yet see them.

In 1829, he submitted work on equations to the Academy of Sciences. Augustin-Louis Cauchy examined part of the material. For many years, accounts claimed that Cauchy lost or rejected the manuscripts, but the surviving documentation suggests a more nuanced story: he appears to have recognized their interest and recommended that they be reorganized for an Academy prize.

Galois prepared a new memoir and submitted it in 1830 to Joseph Fourier, the institution’s secretary. Fourier died shortly afterward, and the manuscript was not among the papers considered for the prize.

In 1831, Galois submitted another version. Siméon Denis Poisson found it insufficiently clear and developed to permit an evaluation of its rigor (Rothman, 1982; Toti Rigatelli, 1996). The recommendation to expand the exposition was not entirely unjustified, but to Galois it represented another closed door.

He possessed extraordinary discoveries but could not make them pass through the mechanisms responsible for recognizing them.

Mathematics in the Middle of a Revolution

Galois lived during a time of immense political unrest. The July Revolution of 1830 removed Charles X from the throne, but the new regime of Louis Philippe disappointed republicans who had hoped to establish a republic.

Galois became increasingly radical. He publicly criticized his school’s director for preventing students from participating in the revolutionary events and was expelled.

He joined the Artillery of the National Guard, a unit with a strong republican presence that the government later dissolved. He attended political meetings, wrote articles, and associated with militants who regarded the new monarchy as inadequate.

In May 1831, he was arrested after making a toast that the authorities interpreted as a threat against the king. A jury acquitted him. Two months later, he was arrested again for carrying weapons and wearing the uniform of the dissolved unit. This time, he received a prison sentence.

He continued revising his mathematics while incarcerated.

His genius did not exist separately from his historical moment. Equations, academic frustration, republicanism, and imprisonment belonged to the same life, lived with enormous intensity and little patience.

A Duel of Uncertain Cause

Galois left prison in April 1832. Soon afterward, he became involved in a confusing relationship with Stéphanie-Félicie Poterin du Motel, the daughter of a physician associated with the residence where he was staying.

On May 30, he fought a pistol duel against an opponent whose identity has never been established conclusively (Rothman, 1982).

Some accounts connect the conflict to Stéphanie. Others propose a quarrel between republicans or even a political provocation. None of these explanations possesses enough evidence to be considered definitive.

Galois himself wrote that he was dying as the victim of “an infamous coquette and her two dupes,” but this statement does not allow the events to be reconstructed with certainty.

The story of a genius murdered in a political conspiracy is compelling. So is the story of a young man destroyed by impossible love. The surviving documentation does not permit us to choose confidently between them.

What is certain is that Galois believed he might die and decided to organize his ideas before the encounter.

The Night That Became a Legend

During the night of May 29, Galois wrote several letters. The most important was addressed to his friend Auguste Chevalier. In it, he described results involving equations, integrals, and groups, and asked that his work be sent to prominent mathematicians.

He also reviewed earlier manuscripts, added comments, and identified arguments that required further development.

According to the legend, he created all of group theory in a single night. That did not happen. His publications and drafts demonstrate that he had been developing these ideas for several years (Galois, 2011; Rothman, 1982).

What is extraordinary is not that he invented an entire branch of mathematics within a few hours. It is that, believing death to be near, he understood which of his results were truly essential and urgently attempted to communicate them.

The letter was less a sudden act of creation than an intellectual testament.

A Gunshot and Hours of Abandonment

The duel took place on the morning of May 30, 1832. Galois was shot in the abdomen. His opponent left the scene, and the wounded young man remained there until he was found and taken to a hospital.

His brother Alfred came to be with him.

Évariste Galois died on May 31 at the age of twenty. He was buried on June 2 in a common grave at Montparnasse Cemetery. The exact location of his burial is unknown.

Only days later, a republican uprising erupted in Paris. Some of his political companions attended the funeral, and the atmosphere was tense enough to raise fears that the ceremony might become a demonstration.

The mathematician and the revolutionary disappeared together before France understood the full significance of either one.

The Pages That Waited Fourteen Years

Auguste Chevalier and Alfred Galois preserved the manuscripts. Recognition, however, did not come immediately.

In 1843, mathematician Joseph Liouville announced that he had examined Galois’s writings and understood their importance. In 1846, he published a fundamental portion of them in the Journal de mathématiques pures et appliquées (Galois, 2011; University of St Andrews, n.d.).

Fourteen years had passed since the duel.

Galois’s ideas gradually became the foundation of Galois theory and contributed to the development of group theory. Their influence later extended into geometry, number theory, physics, and cryptography.

Whenever the symmetries of a structure are studied, whenever a transformation is analyzed not in isolation but as part of a system, something remains of the change in perspective he initiated.

The Danger of a Story That Is Too Perfect

Galois’s life appears designed to become a myth: a misunderstood teenager, institutions rejecting his manuscripts, revolutions, imprisonment, a mysterious love affair, a duel, and brilliant pages written before dawn.

That is precisely why it must be told carefully.

Not all his teachers failed to recognize his talent. Not all his manuscripts were destroyed through negligence. He did not invent his entire theory during his final night. Nor do we know with certainty who arranged the duel or why.

The reality remains extraordinary without embellishment.

Évariste Galois discovered that equations concealed symmetries. His own life, however, was dominated by forces he never managed to organize: politics, institutions, impatience, and an absurd confrontation that killed him before his twenty-first birthday.

His contemporaries saw a troublesome student and a dangerous republican.

Later generations found in his pages a new language for describing the hidden order of mathematics.