Évariste Galois was a French Mathematician who is often regarded as the father of algebra. Galois led a colourful, yet tragic life that ended at the age of 20. His major contribution to mathematics was the discovery of when it is possible to solve an algebraic equation by radicals. His examination of the structure of roots was also of great importance for our understanding of the role of symmetry in mathematical equations. Galois managed to publish some of his work before his untimely death although his achievements were only really noticed decades later. He was also very involved in French republican politics yet his interest in social justice played a significant role in his fall from grace.
Early life and education
He was born in Bourg-la-Rein, a town located to the south of Paris in 1811. Galois’ father’s name was Nicholas Gabriel Galois; his mother was called Adelaide Marie Demante. Both his parents were well educated in areas such as philosophy and literature but, curiously, did not seem to show any particular interest or aptitude for mathematics. Adelaide taught Galois a range of disciplines, including Latin, Greek and religion, herself until he reached the age of 12. His fatherwas a highly influential republican politician and he was elected mayor of the town in 1815.
Galois did not fit into a conventional academic mould, he had an original and creative mind, but although he did well in many subjects at school, winning prizes, he sometimes struggled with conventional examinations and forms of assessments. His literature teacher commented: “he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly, for, after his examination, I believed him to have but little intelligence”. This conclusion was undoubtedly reached due to the fact that, towards the end of his school career, Galois became increasingly obsessed with mathematics at the expense of other subjects.Galois was fascinated by Adrien Marie Legendre’s Elements of Geometry and the work ofJoseph-Louis Lagrange. However, his self-absorbed style of study meant that he often neglected the textbooks required for formal assessment. It was evident at this stage and throughout his very short life that people simply did not understand him, partly due to a difficulty in expressing his ideas in a comprehensible manner. His mathematics examiner did recognise his talent and stated: “This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research”.
Galois sought to enter the École Polytechnique in 1828 as it was the most prestigious and elite institution for mathematics in France. It was also a highly politicised hotbed of republicanism. He failed to gain entry as he was not sufficiently prepared for the examination. Galois then made a second attempt to gain entry and failed once more. One story suggests that, in a fit of angry frustration he threw an eraser at the examiner. Subsequently, Galois had to accept that he was not going to get into his favoured institution; he took up the opportunity to attend the less prestigious École Normale instead. During unrest, Galois and his contemporaries were forcibly kept in the institution under lock and key. Galois made a complaint by writing a letter to the press and was expelled for his actions.
The failure of his second attempt to gain entry into theÉcole Polytechnique was influenced by the death of his father in July 1829. His father had committed suicide following a series of rumours that were being spread by his political opponents. During the funeral, Galois confronted the town’s priest who he believed had started the rumours. Those gathered at the funeral chased the priest out of the church grounds.
Recognition was very important to Galois: he sent his work to leading French mathematician Augustin Louis Cauchy who told him that it had similarities to the work of Norwegian mathematician Niels Abel. This led him to study Abel’s work in more detail while developing his own ideas. He soon submitted his work to Joseph Fourier at the French Academy in the hope that he might be considered for the Grand Prize. Unfortunately for Galois, the prize was given to Abel and Jacobi.
However, Galois did achieve some success too. Most notably, he was able to publish several of his papers. In April 1830, Bulletin des Sciences Mathématiques(The Bulletin of Mathematical Sciences) published a shortened version of the paper that he had submitted for the Grand Prize on the algebraic approach to solving equations. The same journal published more of his work in June of that year on the resolution of numerical equations. The Bulletin published a third paper on the theory of numbers also in June 1830. He published his fourth and final paper in December of that year which dealt with a few points of mathematical analysis.
Where did Galois’ political radicalism come from? His family background clearly played a key role. Nonetheless, the condition of France during his lifetime must have also had an influence on his views. The France of the early nineteenth century had many problems. Napoleon had been defeated by the allies and, following his removal, was replaced by the reactionary King Charles X. Towns and cities were growing with rapid industrialisation, but this was accompanied by impoverished living conditions in urban slums. In 1830 unemployment in the main cities, such as Paris, escalated with thousands of people unable to find work. Charles X soon had to flee the country.Galois lived in a time of great political unrest and a spirit of rebellion was very much the order of the day among educated young people. Even Galois’ school theLycée of Louis-le-Grand was characterised by rebellious students.
In May 1831, 200 republicans gathered for a dinner in celebration of the acquittal of 19 officers who had been accused of an attempt to overthrow the government. Galois stood up at the dinner to raise his wine glass in one hand and a dagger in the other. He was reported to have made threats to King Louis Phillipe. Immediately after the dinner, Galois was arrested and taken to Sainte-Pélagie prison.On this occasion, his lawyer was able to get him released without charge as he claimed that Galois’s words had been misinterpreted due to the excessive noise in the room. However, while he was now being monitored very closely, Galois refused to compromise his political beliefs by keeping a low profile. He was soon arrested a second time shortly after the first incident because he wore the banned uniform of the National Guard and was found with weapons. This time the French authorities were less lenient and he was sentenced to nine months in prison.
In prison Galois’ life went from bad to worse. He received news that one of his mathematical works had been rejected for publication. Galois tried to kill himself with a dagger, but fellow prisoners managed to stop him from doing so. In March 1832, Galois and some of his fellow prisoners were moved to the pension Sieur Faultrier due to a cholera outbreak in Paris. In the pension, Galois fell in love with Stephanie-Felice du Motel who was the daughter of the physician working on the premises. When Galois was released, he attempted to develop a relationship with Stephanie, but it soon became clear that the feeling was not mutual.
Galois was challenged to a duel with Perscheux d’Herbinville who was an acquaintance from the republican movement. We still do not know for sure what the reason was for the duel. However, a newspaper reported the event at the time, suggesting that it was a conflict between two friends over a female (almost certainly Stephanie). It is possible that Galois had a premonition of imminent death after he had been challenged to the duel. The night before his impending confrontation, he wrote a letter to a friend to explain his important mathematical discoveries. In the early hours of the 30th of May 1832, he was shot. He was discovered by a peasant as his fellow duellist d’Herbinville had left him for dead. He did not die at the scene, but his wound was so severe that he passed away on the following day in Cochin hospital. His funeral took place on 2 June and it formed part of a republican rally which was followed by a number of riots.
After Galois’ death, his brother and friends copied out his work and then sent it to leading mathematicians of the time. Despite their efforts, there was no response. Several years later, Joseph Liouville, another French mathematician, saw the importance in Galois’ papers. Liouville had the papers published in his own mathematics journal. Only then did the significance of Galois’ work become known and understood by fellow mathematicians.
Galois was interested in solutions to polynomial equations. He made many important contributions to group theory. His was able to provide an answer to a question that mathematicians had been struggling with for hundreds of years: establishing when it is possible to solve an algebraic equation by radicals.The smallest radical is essentially the square root which is depicted by the symbol √.Galois discovered that if we take the highest power in a given equation and that power is less than 5, then it is possible to solve that equation using radicals. On the other hand, he found that those equations where the highest power is 5 or more are not solvable using radicals. What was original about Galois approach was that he did not take the root structure of equations themselves for granted as other mathematicians had for so long. Having found inspiration in the work of other mathematicians of the day, he began to focus on the roots of equations and the way they were constructed.
What Galois observed was the fact that many equations have solutions that were symmetrical. So a straightforward example of this would be to say that if x2is 4 then x must equal 2. Another symmetrical solution would be to state thatx can also be equal to – 2. Galois was interested in the relationships between these different symmetries. Symmetry is essential to our understanding of ourselves and the world around us. From our earliest years, our whole logic is structured around symmetries in our efforts to neatly structure objects and our sense of reality. Symmetry is often associated with beauty in the human world and beyond. Symmetry can also be extended to the very structure of the universe itself. Galois did not simply resolve an arcane intellectual mathematical problem, his breakthrough is of great importance to understanding that structure. In essence, his big idea was that certain polynomial equations can be solved by looking at their symmetries. This approach allowed Galois to show why some equations can be solved while others cannot be solved.
Following Galois’ death, many of his followers took up his idea to investigate the many different types of symmetry in mathematical equations. However, Galois’ legacy has proved to be even more important. Physicists have used Galois’ methods as an approach to understanding the very substance of our entire universe. It helped describe existing particles in the universe and more recently it has inspired the search for other constituent elements of matter. In particle physics, the hadron collider was constructed near Geneva in Switzerland between 1998 and 2008 by the European Organization for Nuclear Research (CERN). This is one of the world’s most astonishing experiments to test for the existence of new particles, according to the symmetrical theories of particle physicists. These theories were directly drawn from the ideas of Galois. So although Galois’ life was marked by constant struggles to be understood and to have his original ideas recognised, his legacy has proved to be of great importance for our understanding of mathematics and particle physics.