Maths, revolution, and a femme fatale: The short life of Évariste Galois

Paris, 1832. In the early hours of the 29th of May, a young Frenchman wrote two letters. The first, a note to his friends covered in scribbles, crossed out names of love and phrases of delirium: “je n’ai pas le temps” (“I don’t have the time”). The other, a paper containing fascinating mathematical discoveries that would ignite a generation of thought and, as mathematician Arthur Cayley would later put it, "marked an epoch in the progress of the theory of algebraic equations." The next day, young Évariste Galois would be found bleeding out, a bullet in his abdomen. He died a day later, aged just 20.  

Galois’ Youth and Education  

Galois was born in Bourg-la-Reine, near Paris, in 1811 to an upper-class family. Little is known about his childhood although it seems it was a happy one. At 12 years old, he entered the preparatory school Lycée de Louis-le-Grand. It was here that his brilliance was recognised, and he began to take a serious interest in mathematics. He stumbled across a copy of Adrien-Marie Legendre's Éléments de Géométrie, a classic textbook seen as a rigorous alternative to Euclid's Elements. It was said that Galois read the book like a novel and mastered it in one reading. At Louis-le-Grand, he caught his first glimpse of revolutionary politics: in his first year, students held a rebellion and refused to sing in chapel. This resulted in the expulsion of hundreds of students, and it was the influence of this environment that would govern Galois’ approach to politics later in life.  

Despite his ability, Galois was an extremely untidy worker, causing him to fail the entrance exam for the École Polytechnique, albeit a year early. Instead, he entered the École Préparatoire, considered far below the standard of the Polytechnique. The next year, Galois published his first research paper, although it was considered insubstantial and held no hint of genius. At the same time, he was starting to make fundamental discoveries about polynomial equations and submitted these to the Academy of Sciences. The referee, the celebrated mathematician Augustin-Louis Cauchy, refused to accept them for publication, the reasons for which remain a mystery. Nonetheless, it is widely believed that Cauchy was generally impressed with Galois' work and recognised the merits of the papers, which would form the foundations of his legacy.  

Politics  

In the summer of 1829, Galois' father committed suicide following a bitter political dispute with his local priest, who had forged his father's signature on malicious epigrams directed at his own relatives. A few days later, Galois sat the entrance exam for the École Polytechnique and failed again. Legend has it that he lost his temper and launched an eraser at the examiner, speaking to his emotional and volatile character.  

The academic life of Galois was shrouded by the political turmoil of the French Revolution. In 1830, King Charles X faced abdication, so to maintain power he published the notorious July Ordinances, which suspended the freedom of the press. This led to the July Revolution, an uprising of the people which lasted three days, and ended with the Duke of Orléans Louis-Phillipe on the throne. During this revolution, Galois and his peers had been locked in the École Normale by the school's Director. In his frustration, Galois wrote an attack to the Director in the student newspaper, which resulted in the decision to have him expelled. Nonetheless, before the expulsion could come into place, he quit the École and joined the artillery unit of the National Guard, a strongly Republican unit. He divided his time between mathematics and politics until the end of the year, when the King disbanded the artillery unit out of fears they would destabilise the government.  

In April 1831, nineteen former members of Galois's unit were put on trial for an attempt to overthrow the government; they were acquitted, and a feast was held. During this banquet, Galois raised a toast to Louis-Phillipe, which was interpreted by the Republicans as a direct threat to the king. Galois was arrested the next day but was again acquitted. On Bastille Day 1831, Galois led a Republican demonstration, wearing the uniform of the disbanded artillery unit and equipped with guns and daggers, all of which was illegal. Galois was arrested and put in jail. He spent almost a year there, awaiting appeals. After the cholera epidemic of 1832, Galois was put on parole. The young man’s life was marked by an energetic zeal for reactionary politics, extremely passionate about the Republic.  

Galois’ Final Months  

Towards the end of his life, Galois experienced what was probably his one and only love affair. Who this was with is widely disputed, yet ‘Stéphanie D.’, whose name had been written and scribbled out numerous times in Galois’ manuscripts, appears the most likely identity. Recently, the name has been traced to the daughter of a physician who worked at the hospital where Galois was sent after his release from prison amid the cholera epidemic. The story goes: Galois was rejected by Stéphanie and took the news horribly. Not long after this, he was challenged to a pistol duel as a consequence of his conduct with Stéphanie. Alexandre Dumas, who had attended the banquet with Galois, claimed the challenger was Peschaux D'Herbinville, a fellow Republican and one of the nineteen members of the artillery unit put on trial, amplifying the drama of the ordeal. Another theory is that the duel was a set up by his royalist enemies, to subtly vanquish their political enemy in a foolish affaire d’honneur.  

On the eve of the duel, Galois wrote his famous letters, in which he drew connections between two fundamental mathematical objects, groups and polynomial equations, stating that the structure of the 'Galois group' can determine whether a polynomial equation can be solved. Alongside the maths, he wrote words of delirious love – fervent crossings out just barely covering the words “une femme” (“woman”) several times, painting the picture of a true femme fatale. 

The Theory  

Everyone can remember (or at least remember trying to remember) the quadratic formula. This gave you the solutions to an equation (polynomial) with a term raised to the power of 2 (degree 2). But what about degree 3? Is there a formula for that? What about 4, 5, and so on? Mathematicians in the 16th century, constantly squabbling over credit of the cubic (degree 3) and quartic (degree 4) equations, found them. The quintic (degree 5), for a long time, was an enigma. Galois’ scribbled notes proved that a general formula for the solutions of a polynomial of degree greater than or equal to 5 didn't exist. Galois had solved the unsolvable.  

What Galois achieved in such a short lifespan, crammed with extraordinary setbacks, was remarkable. One must wonder what else he could have discovered had he not died in a passion-fuelled duel. Galois' elegant connection of two seemingly disparate parts of mathematics paved the way for advances in cryptography, physics and coding theory. Despite his tragic life being cut short, his work will guide mathematicians for centuries to come. 

Bibliography 

Rothman, Tony, “Genius and Biographers: The Fictionalization of Evariste Galois”, The American Mathematical Monthly 89, no. 2 (1982), pp. 84–106, https://doi.org/10.2307/2320923.  

Stewart, Ian, Galois Theory, 3^rd edn (CRC Press, 2003).