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MOST IMPORTANT SCIENTIFIC DISCOVERIES
Group theory. History and essence of scientific discovery
Directory / The most important scientific discoveries Permutation groups of roots were dealt with earlier by Lagrange and Gauss. But the merit of the one who formulated the essential properties of concepts and applied them to the solution of new and difficult problems is indisputable. This was done by the French mathematician Galois for the concept of a group. Only after his work did it become a subject of study for mathematicians. Évariste Galois (1811–1832) was born in Bourg-la-Reine. In 1823, Evariste was sent by his parents to study at the Royal College in Paris. Here he became interested in mathematics and began to independently study the works of Legendre, Euler, Lagrange, Gauss. Lagrange's ideas completely take over Galois. It seems to him, as once to Abel, that he has found a solution to the equation of the fifth degree. He makes an unsuccessful attempt to enter the Polytechnic School, but the knowledge of the works of Legendre and Lagrange was not enough, and Galois returned to college. Here happiness smiles for the first time - he meets a teacher who was able to appreciate his genius. Richard knew how to rise above the official programs, he was aware of the progress of the sciences and sought to broaden the horizons of his students. Richard's comments about Evariste are simple: "He works only in the higher areas of mathematics." Indeed, already at the age of seventeen, Galois received the first scientific results. In 1829, his note "Proof of a Theorem on Periodic Continued Fractions" was published. At the same time, Galois presented another work to the Paris Academy of Sciences. She got lost at Kosha's. Galois tries to re-enter the Polytechnic School, and again fails. To this was soon added an event that shocked the young man: hunted by political opponents, his father committed suicide. The misfortunes that befell Evariste inevitably affected him: he became nervous and quick-tempered. In 1829 Galois entered the Normal School. It prepared candidates for the title of teacher. Here Evarist completed a study on the theory of algebraic equations and in 1830 submitted his work to the competition of the Paris Academy of Sciences. His fate was in the hands of the permanent secretary of the Academy - Fourier. Fourier begins to read the manuscript, but soon dies. The second manuscript, like the first, disappears. In the life of Galois, a time has come filled with important events. He joined the Republicans, joined the "Society of Friends of the People" and enlisted in the artillery of the National Guard. For speaking out against the leadership, he was expelled from the Normal School. On July 14, 1831, in commemoration of the next anniversary of the storming of the Bastille, a manifestation of the Republicans took place. The police arrested many demonstrators, among them was Galois. The trial of Galois took place on October 23, 1831. He was sentenced to 9 months in prison. Galois continued his research in prison. On the morning of May 30, 1832, in a duel in the town of Gentilly, Galois was mortally wounded by a bullet in the stomach. He died a day later. Galois's mathematical works, at least those that survive, are sixty small pages long. Never before has a work of such a small volume brought such wide fame to the author. In 1832, Galois, while in prison, draws up a program that was published only seventy years after his death. But even at the beginning of the twentieth century, it did not arouse serious interest and was soon forgotten. Only modern mathematicians, who continued the work of many generations of scientists, finally realized Galois's dream. "I beg my judges to at least read these few pages," Galois began his famous memoir. However, Galois's ideas were so deep and comprehensive that at that time it was really difficult for any scientist to appreciate them. "... So, I believe that the simplifications obtained by improving calculations (of course, we mean fundamental simplifications, not technical ones) are not at all unlimited. The moment will come when mathematicians will be able to foresee algebraic transformations so clearly, that the expenditure of time and paper in carrying them out carefully will cease to pay off.I do not claim that analysis cannot achieve anything new beyond such a foresight, but I think that without it all means will one day be in vain. To subordinate calculations to one's will, to group mathematical operations, to learn to classify them according to the degree of difficulty, and not according to external signs - these are the tasks of the mathematicians of the future as I understand them, this is the path I want to take. Let no one confuse the vehemence I have shown with the desire of some mathematicians to avoid any calculations at all. Instead of algebraic formulas, they use lengthy arguments and, to the cumbersomeness of mathematical transformations, they add the cumbersomeness of a verbal description of these transformations, using a language that is not adapted to perform such tasks. These mathematicians are a hundred years behind. Nothing of the kind happens here. Here I am doing analysis analysis. At the same time, the most complex transformations now known (elliptic functions) are considered only as special cases, very useful and even necessary, but still not general, so that refusing further broader research would be a fatal mistake. The time will come when the transformations referred to in the higher analysis outlined here will actually be carried out and will be classified according to the degree of difficulty, and not according to the type of functions arising here. Here it is necessary to pay attention to the words "group mathematical operations". Galois undoubtedly means by this the theory of groups. First of all, Galois was not interested in individual mathematical problems, but in general ideas that determine the entire chain of considerations and guide the logical course of thought. His evidence is based on a deep theory that allows you to combine all the results achieved by that time and determine the development of science for a long time to come. A few decades after Galois's death, the German mathematician David Hilbert called this theory "the establishment of a certain framework of concepts." But no matter what name it is given, it is obvious that it covers a very large area of knowledge. "In mathematics, as in any other science," Galois wrote, "there are questions that need to be addressed at this very moment. These are the pressing problems that capture the minds of advanced thinkers, regardless of their own will and consciousness." One of the problems that Évariste Galois worked on was the solution of algebraic equations. What happens if we consider only equations with numerical coefficients? After all, it may happen that although there is no general formula for solving such equations, the roots of each individual equation can be expressed in radicals. What if it's not? Then there must be some sign that allows you to determine whether this equation is solved in radicals or not? What is this sign? The first of Galois's discoveries was that he reduced the degree of uncertainty of their meanings, that is, he established some of the "properties" of these roots. The second discovery is related to the method used by Galois to obtain this result. Instead of studying the equation itself, Galois studied its "group", or, figuratively speaking, its "family". “A group,” writes A. Dalma, “is a collection of objects that have certain common properties. Let, for example, real numbers be taken as such objects. The general property of a group of real numbers is that when multiplying any two elements of this group, we get is also a real number. Instead of real numbers, motions on the plane, studied in geometry, can appear as "objects"; in this case, the property of the group is that the sum of any two motions again gives motion. Passing from simple examples to more complex ones, one can, as "objects" to choose some operations on objects. In this case, the main property of the group will be that the composition of any two operations is also an operation. It was this case that Galois studied. Considering the equation that needed to be solved, he associated with it a certain group of operations (to Unfortunately, we are unable to clarify here how this is done) and proved that the properties of the equationreflect the characteristics of this group. Since different equations may have the same group, it suffices to consider the group corresponding to them instead of these equations. This discovery marked the beginning of the modern stage in the development of mathematics. No matter what "objects" the group consists of: numbers, movements or operations, they can all be considered as abstract elements that do not have any specific features. In order to define a group, it is only necessary to formulate the general rules that must be followed in order for a given set of "objects" to be called a group. At present, mathematicians call such rules group axioms, group theory consists in listing all the logical consequences of these axioms. At the same time, more and more new properties are consistently discovered; proving them, the mathematician more and more deepens the theory. It is essential that neither the objects themselves nor the operations on them are specified in any way. If after this, in the study of some particular problem, one has to consider some special mathematical or physical objects that form a group, then, based on the general theory, one can foresee their properties. The theory of groups, therefore, provides tangible savings in funds; in addition, it opens up new possibilities for the application of mathematics in research work. The introduction of the concept of a group saved mathematicians from the burdensome duty of considering many different theories. It turned out that it was only necessary to single out the "basic features" of one theory or another, and since, in fact, they are all completely similar, it is enough to designate them with the same word, and it immediately becomes clear that it is pointless to study them separately. Galois seeks to introduce a new unity into the overgrown mathematical apparatus. Group theory is, first of all, putting things in order in mathematical language. Group theory, starting from the end of the XNUMXth century, had a huge impact on the development of mathematical analysis, geometry, mechanics and, finally, physics. It subsequently penetrated into other areas of mathematics - Lie groups appeared in the theory of differential equations, Klein groups in geometry. There also arose Galileo groups in mechanics and the groups Lorenz in the theory of relativity. Author: Samin D.K.
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