Fermat's Last Theorem

by Simon Singh

  On 4 December the thwarted genius attempted to become a professional rebel by joining the Artillery of the National Guard, a republican branch of the militia otherwise known as the ‘Friends of the People’. Before the end of the month the new king Louis-Phillipe, anxious to avoid a further rebellion, abolished the Artillery of the National Guard, and Galois was left destitute and homeless. The most brilliant young talent in all of Paris was being persecuted at every turn and some of his former mathematical colleagues were becoming increasingly worried about his plight. Sophie Germain, who was by this time the shy elder stateswoman of French mathematics, expressed her concerns to friend of the family Count Libri-Carrucci:

  Decidedly there is a misfortune concerning all that touches upon mathematics. The death of Monsieur Fourier has been the final blow for this student Galois who, in spite of his impertinence, showed signs of a clever disposition. He has been expelled from the Ecole Normale, he is without money, his mother has very little also and he continues his habit of insult. They say he will go completely mad. I fear this is true.

  As long as Galois’s passion for politics continued it was inevitable that his fortunes would deteriorate further, a fact documented by the great French writer Alexandre Dumas. Dumas was at the restaurant Vendanges de Bourgogne when he happened upon a celebration banquet in honour of nineteen republicans aquitted of conspiracy charges:

  Suddenly, in the midst of a private conversation which I was carrying on with the person on my left, the name Louis-Phillipe, followed by five or six whistles, caught my ear. I turned around. One of the most animated scenes was taking place fifteen or twenty seats from me. It would be difficult to find in all Paris two hundred persons more hostile to the government than those to be found reunited at five o’clock in the afternoon in the long hall on the ground floor above the garden.

  A young man who had raised his glass and held an open dagger in the same hand was trying to make himself heard – Evariste Galois was one of the most ardent republicans. The noise was such that the very reason for this noise had become incomprehensible. All that I could perceive was that there was a threat and that the name of Louis-Phillipe had been mentioned: the intention was made clear by the open knife.

  This went way beyond my own republican opinions. I yielded to the pressure from my neighbour on the left who, as one of the King’s comedians, didn’t care to be compromised, and we jumped from the window sill into the garden. I went home somewhat worried. It was clear this episode would have its consequences. Indeed, two or three days later, Evariste Galois was arrested.

  After being detained at Sainte-Pélagie prison for a month Galois was charged with threatening the King’s life and brought to trial. Although there was little doubt from his actions that Galois was guilty, the raucous nature of the banquet meant that nobody could actually confirm that they had heard him make any direct threats. A sympathetic jury and the rebel’s tender age – he was still only twenty – led to his acquittal. The following month he was arrested again.

  On Bastille Day, 14 July 1831, Galois marched through Paris dressed in the uniform of the outlawed Artillery Guard. Although this was merely a gesture of defiance, he was sentenced to six months in prison and returned to Sainte-Pélagie. During the following months the teetotal youth was driven to drink by the rogues who surrounded him. The botanist and ardent republican François Raspail, who was imprisoned for refusing to accept the Cross of the Legion of Honour from Louis-Phillipe, wrote an account of Galois’s first drinking bout:

  He grasps the little glass like Socrates courageously taking the hemlock; he swallows it as one gulp, not without blinking and making a wry face. A second glass is not harder to empty than the first, and then the third. The beginner loses his equilibrium. Triumph! Homage to the Bacchus of the jail! You have intoxicated an ingenuous soul, who holds wine in horror.

  A week later a sniper in a garret opposite the prison fired a shot into a cell wounding the man next to Galois. Galois was convinced that the bullet was intended for himself and that there was a government plot to assassinate him. The fear of political persecution terrorised him, and the isolation from his friends and family and rejection of his mathematical ideas plunged him into a state of depression. In a bout of drunken delirium he tried to stab himself to death, but Raspail and others managed to restrain and disarm him. Raspail recalls Galois’s words immediately prior to the suicide attempt:

  Do you know what I lack my friend? I confide it only to you: it is someone I can love and love only in spirit. I have lost my father and no one has ever replaced him, do you hear me …?

  In March 1832, a month before Galois’s sentence was due to finish, a cholera epidemic broke out in Paris and the prisoners of Sainte-Pélagie were released. What happened to Galois over the next few weeks has been the subject of intense speculation, but what is certain is that the events of this period were largely the consequence of a romance with a mysterious woman by the name of Stéphanie-Félicie Poterine du Motel, the daughter of a respected Parisian physician. Although there are no clues as to how the affair started, the details of its tragic end are well documented.

  Stéphanie was already engaged to a gentleman by the name of Pescheux d’Herbinville, who uncovered his fiancée’s infidelity. D’Herbinville was furious and, being one the finest shots in France, he had no hesitation in immediately challenging Galois to a duel at dawn. Galois was well aware of his challenger’s reputation. During the evening prior to the confrontation, which he believed would be his last opportunity to commit his thoughts to paper, he wrote letters to his friends explaining his circumstances:

  I beg my patriots, my friends, not to reproach me for dying otherwise than for my country. I died the victim of an infamous coquette and her two dupes. It is in a miserable piece of slander that I end my life. Oh! Why die for something so little, so contemptible? I call on heaven to witness that only under compulsion and force have I yielded to a provocation which I have tried to avert by every means.

  Despite his devotion to the republican cause and his romantic involvement, Galois had always maintained his passion for mathematics and one of his greatest fears was that his research, which had already been rejected by the Academy, would be lost forever. In a desperate attempt to gain recognition he worked through the night writing out the theorems which he believed fully explained the riddle of quintic equations. The pages were largely a transcription of the ideas he had already submitted to Cauchy and Fourier, but hidden within the complex algebra were occasional references to ‘Stéphanie’ or ‘une femme’ and exclamations of despair – ‘I have not time, I have not time!’ At the end of the night, when his calculations were complete, he wrote a covering letter to his friend Auguste Chevalier, requesting that, should he die, the papers be distributed to the greatest mathematicians in Europe:

  My Dear Friend,

  I have made some new discoveries in analysis. The first concern the theory of quintic equations, and others integral functions.

  In the theory of equations I have researched the conditions for the solvability of equations by radicals; this has given me the occasion to deepen this theory and describe all the transformations possible on an equation even though it is not solvable by radicals. All this will be found here in three memoirs …

  In my life I have often dared to advance propositions about which I was not sure. But all I have written down here has been clear in my head for over a year, and it would not be in my interest to leave myself open to the suspicion that I announce theorems of which I do not have a complete proof.

  Make a public request of Jacobi or Gauss to give their opinions, not as to the truth, but as to the importance of these theorems. After that, I hope some men will find it profitable to sort out this mess.

  I embrace you with effusion,

  E. Galois

  The following morning, Wednesday 30 May 1832, in an isolated field Galois and d’Herbinville faced each other at twenty-five paces armed with pistols. D’Herbinville was accompanied
by seconds; Galois stood alone. He had told nobody of his plight: a messenger he had sent to his brother Alfred would not deliver the news of the duel until it was over and the letters he had written the previous night would not reach his friends for several days.

  The pistols were raised and fired. D’Herbinville still stood, Galois was hit in the stomach. He lay helpless on the ground. There was no surgeon to hand and the victor calmly walked away leaving his wounded opponent to die. Some hours later Alfred arrived on the scene and carried his brother to Cochin hospital. It was too late, peritonitis had set in, and the following day Galois died.

  His funeral was almost as farcical as his father’s. The police believed that it would be the focus of a political rally and arrested thirty comrades the previous night. Nonetheless two thousand republicans gathered for the service and inevitably scuffles broke out between Galois’s colleagues and the government officials who had arrived to monitor events.

  The mourners were angry because of a growing belief that d’Herbinville was not a cuckolded fiancé but rather a government agent, and that Stéphanie was not just a lover but a scheming seductress. Events such as the shot which was fired at Galois while he was in Sainte-Pélagie prison already hinted at a conspiracy to assassinate the young trouble-maker, and therefore his friends concluded that he had been duped into a romance which was part of a political plot contrived to kill him. Historians have argued about whether the duel was the result of a tragic love affair or politically motivated, but either way one of the world’s greatest mathematicians was killed at the age of twenty, having studied mathematics for only five years.

  Before distributing Galois’s papers his brother and Auguste Chevalier rewrote them in order to clarify and expand the explanations. Galois’s habit of explaining his ideas hastily and inadequately was no doubt exacerbated by the fact that he had only a single night to outline years of research. Although they dutifully sent copies of the manuscript to Carl Gauss, Carl Jacobi and others, there was no acknowledgment of Galois’s work for over a decade, until a copy reached Joseph Liouville in 1846. Liouville recognised the spark of genius in the calculation and spent months trying to interpret its meaning. Eventually he edited the papers and published them in his prestigious Journal de Mathématiques pures et appliquées. The response from other mathematicians was immediate and impressive because Galois had indeed formulated a complete understanding of how one could go about finding solutions to quintic equations. First Galois had classified all quintics into two types: those that were soluble and those that were not. Then, for those that were soluble, he devised a recipe for finding the solutions to the equations. Moreover, Galois examined equations of higher order than the quintic, those containing x6, x7, and so on, and could identify which of these were soluble. It was one of the masterpieces of nineteenth-century mathematics created by one of its most tragic heroes.

  In his introduction to the paper Liouville reflected on why the young mathematician had been rejected by his seniors and how his own efforts had resurrected Galois:

  An exaggerated desire for conciseness was the cause of this defect which one should strive above all else to avoid when treating the abstract and mysterious matters of pure Algebra. Clarity is, indeed, all the more necessary when one essays to lead the reader farther from the beaten path and into wilder territory. As Descartes said, ‘When transcendental questions are under discussion be transcendentally clear.’ Too often Galois neglected this precept; and we can understand how illustrious mathematicians may have judged it proper to try, by the harshness of their sage advice, to turn a beginner, full of genius but inexperienced, back on the right road. The author they censured was before them ardent, active; he could profit by their advice.

  But now everything is changed. Galois is no more! Let us not indulge in useless criticisms; let us leave the defects there and look at the merits …

  My zeal was well rewarded, and I experienced an intense pleasure at the moment when, having filled in some slight gaps, I saw the complete correctness of the method by which Galois proves, in particular, this beautiful theorem.

  Toppling the First Domino

  At the heart of Galois’s calculations was a concept known as group theory, an idea which he had developed into a powerful tool capable of cracking previously insoluble problems. Mathematically, a group is a set of elements which can be combined together using some operation, such as addition or multiplication, and which satisfy certain conditions. An important defining property of a group is that, when any two of its elements are combined using the operation, the result is another element in the group. The group is said to be closed under that operation.

  For example, positive and negative whole numbers form a group under the operation of ‘addition’. Combining one whole number with another under the operation of addition leads to a third whole number, e.g.

  Mathematicians state that ‘positive and negative whole numbers are closed under addition and form a group’. On the other hand the whole numbers do not form a group under the operation of ‘division’, because dividing one whole number by another does not necessarily lead to another whole number, e.g.

  The fraction 1⁄3 is not a whole number and is outside the original group. However, by considering a larger group which does include fractions, the so-called rational numbers, closure can be re-established: ‘the rational numbers are closed under division’. Having said this, one stills needs to be careful because division by the element zero results in infinity, which leads to various mathematical nightmares. For this reason it is more accurate to state that ‘the rational numbers (excluding zero) are closed under division’. In many ways closure is similar to the concept of completeness described in earlier chapters.

  The whole numbers and the fractions form infinitely large groups, and one might assume that, the larger the group, the more interesting the mathematics it will generate. However, Galois had a ‘less is more’ philosophy, and showed that small carefully constructed groups could exhibit their own special richness. Instead of using the infinite groups, Galois began with a particular equation and constructed his group from the handful of solutions to that equation. It was groups formed from the solutions to quintic equations which allowed Galois to derive his results about these equations. A century and a half later Wiles would use Galois’s work as the foundation for his proof of the Taniyama–Shimura conjecture.

  To prove the Taniyama–Shimura conjecture, mathematicians had to show that every one of the infinite number of elliptic equations could be paired with a modular form. Originally they had attempted to show that the whole DNA for one elliptic equation (the E-series) could be matched with the whole DNA for one modular form (the M-series), and then they would move on to the next elliptic equation. Although this is a perfectly sensible approach, nobody had found a way to repeat this process over and over again for the infinite number of elliptic equations and modular forms.

  Wiles tackled the problem in a radically different way. Instead of trying to match all elements of one E-series and M-series and then moving on to the next E-series and M-series, he tried to match one element of all E-series and M-series and then move on to the next element. In other words each E-series has an infinite list of elements, individual genes which make up the DNA, and Wiles wanted to show that the first gene in every E-series could be matched with the first gene in every M-series. He would then go on to show that the second gene in every E-series could be matched with the second gene in every M-series, and so on.

  In the traditional approach one had an infinite problem, which was that even if you could prove that all of one E-series matched all of one M-series, there were still infinitely many other E-series and M-series to be matched. Wiles’s approach still involved tackling infinity because even if he could prove that the first gene of every E-series was identical to the first gene of every M-series there were still infinitely many other genes to be matched. However, Wiles’s approach had one major advantage over the traditional approach.


  In the old method, once you had proved that the whole of one E-series matched the whole of one M-series, you then had to ask, Which E-series and M-series do I try and match up next? The infinity of E-series and M-series have no natural order and so whichever one is tackled next is a largely arbitrary choice. Crucially, in Wiles’s method, the genes in the E-series do have a natural order, and so having proved that all the first genes match (E1 = M1), the next step is obviously to prove that all the second genes match (E2 = M2), and so on.

  This natural order is exactly what Wiles needed in order to develop an inductive proof. Initially Wiles would have to show that the first element of every E-series could be paired with the first element of every M-series. Then he would have to show that if the first elements could be paired then so could the second elements, and if the second elements could be paired then so could the third elements, and so on. He had to topple the first domino, and then he had to prove that any falling domino would also topple the next one.