A CERTAIN MATHEMATICIAN: EVARISTE GALOIS

There is a place called Champs de Mars in Paris. It is on the left bank of the Seine in the western part of the city. The field has always been the venue for official military parades. International exhibitions were held there from 1867. The ceremony to administer the oath of loyalty to the first French Constitution was also held here, on 14 July 1790. A popular uprising started on this field on 17 July 1791.  The Champs de Mars Park was first laid out in 1765-1767 and, initially, cadets and the military held military drills and parades. It is so called after the Roman god of war, Mars.  From 1780, it became a popular place for Parisians to promenade. From this field the Montgolfier brothers' balloon rose into the air for the first time in history, on 27 August 1783.

In 1789, a parade in honour of King Louis XVI was held on the field and, a year later, on 14 July 1790, 300,000 Parisians commemorated the first Bastille Day. On 17 July 1791, a bloody clash took place on the Champs de Mars: guardsmen opened fire on demonstrators who demanded the dethroning of the king. From 1867-1937, World Expos were held on the Champs de Mars. And in 1889, at an exhibition which commemorated the 100th anniversary of the French Revolution, the current symbol of Paris, the Eiffel Tower, was built there. The 318-metre-tall, 7,000-ton tower was designed by French engineer Alexandre Gustave Eiffel. Until 1931, the Eiffel Tower was the tallest building in the world. When Eiffel had completed the huge structure there was no radio or television and the newspapers published articles about the pointessness of the tower. Citizens also agreed that the tower was an ugly addition to the urban landscape. The great French writer Guy de Maupassant called the Eiffel Tower a "skeleton" and even dined in restaurants from which he could not see it. Other prominent men of arts also voiced their protests: "In the name of true taste, in the name of art, in the name of French history, we, writers, painters, architects - people who are in love with the immaculate beauty of Paris, express our protest against the erection, in the very heart of the capital, of the huge, useless and ugly Eiffel Tower." However, time is the best judge, as the saying goes. Seen at first as an ugly giant, the Eiffel Tower eventually became the symbol of Paris and now more than 6 million people visit it every year. In addition, the tower which was thought to be useless in the past is now the broadcasting centre for French TV and radio broadcasts, both nationally and abroad. However, it was always regarded as a technological wonder of its time.

Yet another wonder of the same era, whose name entered the history books in letters of gold, was French mathematician Evariste Galois, whose discoveries perplexed even the most respected mathematicians. The formula for the solution of polynomial equations, using radicals, was found in antiquity. But the world learned only 1,000 years later of the mathematics which made it possible to solve algebraic polynomial equations with one unknown. Before that, the work of Italian mathematicians Niccolo Tartaglia, Gerolamo Cardano and Lodovico Ferrari in the XVI century had made it possible to solve equations of the 3rd and 4th degrees. These scientists paved the way for the future development of algebra. And Evariste Galois changed all previous conceptions of mathematics. At the age of 19, he had already proposed axioms and proved theorems. At that time, he was studying at school and signed his scientific papers "Lyceiste." All the laws discovered by Galois are now used in the broadcasting of digital information using different digital TV channels, although Galois himself could not have dreamed that his work would usher in the television era many years later. But time itself handed down its lesson and proved that everything is rational in the history of the development of human thought and that all things fall into their right place given time. And now the "pointless" Eiffel Tower broadcasts digital TV signals which are created using the theorems of the greatest mathematician of his era, Galois. Galois laid the foundation of modern algebra and determined the necessary and sufficient condition for a polynomial of any degree to be solved by radicals. The Galois theory of groups is now called the "Galois region." The notions of "region," "ideal," and so on played key roles in the theory of encoded stability. Without his theory, it would be impossible to transmit digitized information or create digital television.

Highest area of mathematics

Evariste Galois was born on 25 October 1811, in the village of Bourg-la-Reine near Paris. At 12, he joined the Lycee Louis-le-Grand. He was a mediocre student and at 16 he enrolled in mathematical classes. His fondness for mathematics overshadowed all other subjects, to which he did not pay due attention. Here is what his teacher wrote: "Evariste Galois is interested only in the higher areas of mathematics. He loves the subject madly. By asking surprising questions of his teachers, he puts them in awkward positions." Very soon his desire to master mathematical knowledge led Galois to conclude that he would not be able to get the knowledge he wanted from his teachers. And the young man began to read books by the most prominent scientists of the period. He easily mastered the most difficult subjects. At 17, he published his first scientific paper. Because the young prodigy knew much more than his teachers did, Galois's ideas caused perplexity and were not understood.

Galois was able to do calculations without writing them on paper, which made work more difficult for his teachers and irritated them. He reckoned with no one and was easily angered. Galois was denied admission to one of the prestigious French universities, the Ecole Polytechnique, because, during the oral tests, he chose the shortest way of solving problems and gave no further explanations. One year later, during the oral test, he surprised the examiner with his profound logical deliberations, but when he was again denied admission, the great mathematician threw some rags at the commission members and never again entered the building of the Ecole Polytechnique. After this incident, Galois said that he had been asked childish questions which he considered beneath his dignity to answer. The failure to get through the admission tests did not undermine Galois's belief in his mathematical talent and he continued his research independently.  The achievements of the 17-year-old Galois in mathematics make it possible today to understand very different areas of knowledge - the theory of number, crystallography, the physics of elementary particles. He wanted to find solutions to the equations of the 5th degree, which was one of the most difficult mathematical problems of the period. He made great progress towards finding the solution and sent two memoranda to the Academy with the results of his research. These scientific papers by the young researcher influenced another prominent scientist of the period, Augustin-Louis Cauchy, who found Galois's memoranda important enough to nominate them for the Academy of Sciences award in mathematics. Work nominated for the prize had to meet formal criteria and the memoranda were returned to the author with a request to combine them into a single paper. Galois did so and returned the paper to the Academy Secretary, Joseph Fourier, for submission to the jury. Although Galois's work did not include a ready solution for polynomials of the 5th degree, he proposed a very good idea. Many mathematicians agreed that he was one of the worthiest candidates for the award. However, Galois was not only not awarded, but was not even allowed to take part in the competition and his work was lost, together with the papers from Fourier, who died two weeks before the commission session. Galois himself thought that the Academy had deliberately lost his paper.

Some time later, the Academy refused to accept Galois's new paper "On conditions of solutions to polynomials using radicals," citing as the reason "insufficient clarity of proof," which casts doubts on the seriousness of the paper. This was the opinion of prominent French scholar Simeon-Denis Poisson, who voiced it simply because he failed to grasp Galois's idea although, in that paper, Galois had developed the main tenets of the groups theory for solving problems of the theory of equations which had not been resolved before. All these difficulties led Galois to the conclusion that obstacles were being artificially created to keep him at a distance from the mathematical community.

"Professional" revolutionary

The true importance of Galois's theory was discovered only in1870, many years after the tragic death of its author, when a French mathematician, Camille Jordan, associate member of the St Petersburg Academy of Sciences, wrote his book "Substitutions, or a Treatise on Algebraic Equations." Galois's theory had an important influence on the development of not only algebra, but also of the entirety of mathematics in the XIX century. The ideas and methods of Galois's group theory are used in Earth sciences, modern quantum mechanics and crystallography.

After all his difficulties, Galois abandoned mathematics, took an interest in politics and became a supporter of the republicans. Wishing to become a professional revolutionary, he joined the ranks of the National Guard Artillery. When King Louis-Philippe disbanded the National Guard Artillery out of considerations for his own security, Galois, homeless and without any means, had to live like a vagabond. Deeply involved in politics, Galois paid little attention to anything else, so his situation was bound to worsen, to which the great French writer Alexandre Dumas was witness. A recipient of an invitation to a banquet to celebrate the acquittal of 18 Republicans by the courts, Dumas witnessed a loud speech by a young man who held a glass of wine in one hand and a drawn dagger in the other. The young man was Evariste Galois. When he saw that the situation was getting tense and threats against King Louis-Philippe were becoming increasingly frequent, the great writer and the royal commander who sat next to him decided to leave and jumped out of the window.

After one month's imprisonment, Galois was accused of issuing threats against the king, and he appeared before the court. But none of the witnesses at the banquet admitted that they had heard such threats from Galois, so the judge took into account the youth of the accused and found it possible to release him. But he was arrested again in less than a month. The reason was his wearing of the National Guard Artillery uniform, which had been banned on 14 July 1831, Bastille Day. Sentenced to 6 months of imprisonment, Galois was put into solitary confinement. One month before his release, inmates of Parisian prisons were released because of the spread of cholera in the city. There are reports that in prison Galois entered into a love affair with Stephanie, daughter of a respected Parisian doctor. At that time, Stephanie was engaged to a certain Pescheux  D'Herbinville. When he found out about the affair, the young man sent Galois a letter, challenging him to a duel. Galois knew D'Herbinville well from his school years and knew that he was one of the best shots in France. Despite this, he made no attempt to avoid the duel. In his letters to friends, Galois tried to explain the causes of all his troubles: "I beg patriots, my friends, not to reproach me for dying otherwise than for my country. I die the victim of an infamous coquette and her two dupes. It is in a miserable piece of slander that I end my life."

Early in the morning of 30 May 1832, Galois and D'Herbinville stood before each other at a distance of 25 metres. D'Herbinville came to the duel with two seconds, but Galois was alone. Because they had known each other for a long time, the young men decided it was beneath their dignity to aim and fired when they started to move towards each other. Galois's pistol was not loaded, but his opponent's was. Galois was gravely wounded in the stomach, but his opponent left. Galois's brother arrived soon after the duel and took him to hospital, but the great mathematician could not be saved. Galois died when he was only 22.

When you read Galois's article on the shortcomings of training courses for applicants to the Ecole Polytechnique, you get the impression that the author is talking about today's situation. The article, entitled "Teaching Sciences," reads:  "In the past, teachers taught pupils all the knowledge they needed. Now, however, to train youth to join the Ecole Polytechnique, one or two private tutors are required. Poor children, whither shall they listen to their tutors all day or cram what they just heard?  When will they be able to thoroughly weigh all the information they receive and find time to think through the multitude of chaotically piled-up theorems and unrelated algebraic problems? The misinterpreted theories, loaded with useless propositions, are explained in detail and the most beautiful and simple algebraic theorems remain outside the scope of teaching. Instead, the pupils are supplied with information about the excessively lengthy, and not always correct, operations and proofs of results which are known in advance. The test officials want to ask applicants only the most convoluted questions. In these conditions, the pupil cares nothing for learning, only about passing the test. The pupil has to memorize by heart four answers to every theorem."

All the results of Evariste Galois's research fit into 60 small-format pages. But, thanks to his theories, which even the most prominent scientists of his time were unable to appreciate, Galois, who lived only 22 years, will be remembered forever as one of the mathematical geniuses of Europe.