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MOST IMPORTANT SCIENTIFIC DISCOVERIES
Fundamental theorem of algebra. History and essence of scientific discovery
Directory / The most important scientific discoveries "The fundamental theorem of algebra in the form of a statement: An algebraic equation has as many roots as its degree, stated by Girard and Descartes, - notes in his book "In the world of equations" V.A. Nikiforovsky. - Its formulation, which consists in the fact that an algebraic polynomial with real coefficients is decomposed into a product of real linear and quadratic factors, belongs to d'Alembert and Euler. Euler first reported this in a letter to Nicholas I Bernoulli (1687–1759) dated September 1, 1742. From this it followed that the roots of algebraic equations with real coefficients belong to the field of complex numbers. The first proof of the theorem was undertaken in 1746 by d'Alembert (1717–1783). d'Alembert's proof of the fundamental theorem of algebra was, however, analytic, not algebraic. The French mathematician used the concepts of analysis that had not yet taken shape at that time, such as the power series, the infinitesimal. It is not surprising that the proof of the theorem suffered from errors and was later subjected to devastating criticism. Gaussianand then was forgotten. Euler made a new and significant step in the proof of the fundamental theorem of algebra. Leonhard Euler (1707–1783) was born in Basel. After home schooling, thirteen-year-old Leonard was sent by his father to the University of Basel to study philosophy. Among other subjects, elementary mathematics and astronomy were studied at this faculty, taught by Johann Bernoulli. Bernoulli soon noticed the talent of the young listener and began to study with him separately. After receiving a master's degree in 1723, after delivering a speech in Latin on the philosophy of Descartes and Newton, Leonard, at the request of his father, began to study Oriental languages and theology. But he was increasingly attracted to mathematics. Euler began to visit his teacher's house, and between him and the sons of Johann Bernoulli - Nikolai and Daniel - a friendship arose that played a very important role in Leonard's life. In 1725, the Bernoulli brothers were invited to become members of the St. Petersburg Academy of Sciences. They contributed to the fact that Euler moved to Russia. Euler's discoveries, which, thanks to his lively correspondence, often became known long before publication, make his name more and more widely known. His position at the Academy of Sciences was improving. In 1727, he began work with the rank of adjunct, that is, the junior academician, and in 1731 he became a professor of physics, that is, a full member of the Academy. In 1733 he received the chair of higher mathematics, which was previously held by D. Bernoulli, who returned to Basel that year. The growth of Euler's authority found a peculiar reflection in the letters to him of his teacher Johann Bernoulli. In 1728, Bernoulli turns to "the most learned and gifted young man Leonhard Euler", in 1737 - to "the most famous and witty mathematician", and in 1745 - to "the incomparable Leonhard Euler - the head of mathematicians." In 1736 two volumes of his analytical mechanics appeared. The demand for this book was great. Many articles have been written on various questions of mechanics, but there has not yet been a good treatise on mechanics. In 1738, two parts of an introduction to arithmetic appeared in German, in 1739, a new theory of music. At the end of 1740, power in Russia passed into the hands of the regent Anna Leopoldovna and her entourage. An alarming situation has developed in the capital. At this time, the Prussian king Frederick II decided to revive the founded Leibniz Society of Sciences in Berlin, almost inactive for many years. Through his ambassador in Petersburg, the king invited Euler to Berlin. Euler, believing that "the situation began to appear rather uncertain," accepted the invitation. In Berlin, Euler at first gathered around him a small scientific society, and then was invited to the newly restored Royal Academy of Sciences and appointed dean of the mathematical department. In 1743 he published five of his memoirs, four of them on mathematics. One of these works is remarkable in two respects. It indicates a way of integrating rational fractions by decomposing them into partial fractions, and, in addition, the now usual way of integrating higher-order linear ordinary equations with constant coefficients is outlined. In general, most of Euler's work is devoted to analysis. Euler so simplified and supplemented the whole large sections of the analysis of infinitesimals, integration of functions, the theory of series, differential equations, which had already begun before him, that they acquired approximately the form that remains behind them to a large extent to this day. Euler also started a whole new chapter of analysis, the calculus of variations. This initiative of his was soon picked up by Lagrange, and a new science was formed. Euler's proof of the fundamental theorem of algebra was published in 1751 in the work "Investigations on imaginary roots of equations". Euler performed the most algebraic proof of the theorem. Later, his main ideas were repeated and deepened by other mathematicians. Thus, methods for studying equations were first developed by Lagrange, and then became an integral part of Galois theory. The main theorem was that all the roots of the equation belong to the field of complex numbers. To prove this position, Euler established that any polynomial with real coefficients can be expanded into a product of real linear or quadratic factors. Values of numbers that are not real, “Euler called imaginary,” writes Nikiforovsky, “and pointed out that they are usually considered to be those that give real numbers in pairs in sum and product. Therefore, if there are 2 m imaginary roots, then this will give m real quadratic of factors in the polynomial representation Euler writes: “Therefore, it is said that every equation that cannot be factored into real prime factors always has real factors of the second degree. However, no one, so far as I know, has yet proved the truth of this opinion rigorously enough; I will therefore try to give him a proof that covers all cases without exception." The same concept was held by Lagrange, Laplace and some other followers of Euler. Gauss did not agree with her. Euler formulated three theorems that follow from the properties of continuous functions. 1. An odd degree equation has at least one real root. If there are more than one such roots, then their number is odd. 2. An equation of an even degree either has an even number of real roots, or does not have them at all. 3. An equation of even degree, in which the free term is negative, has at least two real roots of different signs. Following this, Euler proved theorems on the decomposability into linear and quadratic real factors of polynomials with real coefficients... When proving the main theorem, Euler established two properties of algebraic equations: 1) a rational function of the roots of the equation, which takes different values for all possible permutations of the roots A, satisfies an equation of degree A, the coefficients of which are expressed rationally in terms of the coefficients of the given equation; 2) if the rational function of the roots of the equation is invariant (does not change) with respect to permutations of the roots, then it is rationally expressed in terms of the coefficients of the original equation. P.S. Laplace, in lectures on mathematics in 1795, following Euler and Lagrange, admits the factorization of a polynomial. At the same time, Laplace proves that they will be real. Thus, both Euler, and Lagrange, and Laplace built the proof of the fundamental theorem of algebra on the assumption of the existence of a factorization field of a polynomial. A special role in the proofs of the main theorem belongs to the "king of mathematicians" Gauss. Carl Friedrich Gauss was born (1777–1855) in Brunswick. He inherited good health from his father's relatives, and a bright intellect from his mother's relatives. At the age of seven, Karl Friedrich entered the Catherine Folk School. In 1788, Gauss moved to the gymnasium. However, it does not teach mathematics. Classical languages are studied here. Gauss enjoys studying languages and is making such progress that he does not even know what he wants to become - a mathematician or a philologist. Gauss is known at court. In 1791 he was presented to Karl Wilhelm Ferdinand, Duke of Brunswick. The boy visits the palace and entertains the courtiers with the art of counting. Thanks to the patronage of the Duke, Gauss was able to enter the University of Göttingen in October 1795. At first he listens to lectures on philology and almost never attends lectures on mathematics. But this does not mean that he does not study mathematics. In 1795, Gauss embraces a passionate interest in whole numbers. In the autumn of the same year, Gauss moved to Göttingen and literally swallowed the literature that fell into his hands for the first time: the works of Euler and Lagrange. "On March 30, 1796, the day of creative baptism comes for him. - writes F. Klein, - Gauss has been for some time already engaged in grouping roots from unity on the basis of his theory of "primordial" roots. And then one morning, waking up, he suddenly clearly and distinctly realized that the construction of a seventeen-gon follows from his theory ... This event was a turning point in the life of Gauss. He decides to devote himself not to philology, but exclusively to mathematics. " Gauss' work becomes for a long time an unattainable example of a mathematical discovery. One of the creators of non-Euclidean geometry, Janos Bolyai, called it "the most brilliant discovery of our time, or even of all time." Only it was difficult to comprehend this discovery! Thanks to the letters to the homeland of the great Norwegian mathematician Abel, who proved the unsolvability of the equation of the fifth degree in radicals, we know about the difficult path that he went through while studying the theory of Gauss. In 1825, Abel writes from Germany: "Even if Gauss is the greatest genius, he obviously did not strive for everyone to understand this at once ..." Gauss' work inspires Abel to build a theory in which "there are so many wonderful theorems that it simply believe." There is no doubt that Gauss also influenced Galois. Gauss himself retained a touching love for his first discovery for life. On March 30, 1796, the day when the regular seventeen-hexagon was built, Gauss's diary begins - a chronicle of his remarkable discoveries. The next entry in the diary appeared on April 8th. It reported on the proof of the theorem of the quadratic law of reciprocity, which he called "golden". Particular cases of this assertion have been proved Farm, Euler, Lagrange. Euler formulated a general conjecture, the incomplete proof of which was given by Legendre. On April 8, Gauss found a complete proof of Euler's conjecture. However, Gauss did not yet know about the work of his great predecessors. He walked the whole difficult path to the "golden theorem" on his own! Gauss made two great discoveries in just 10 days, a month before he turned 19! One of the most surprising aspects of the “Gauss phenomenon” is that in his first works he practically did not rely on the achievements of his predecessors, rediscovering in a short time what had been done in number theory in a century and a half by the works of the greatest mathematicians. In 1801, the famous "Arithmetical Investigations" by Gauss came out. This huge book (more than 500 large format pages) contains the main results of Gauss. "Arithmetic Studies" had a huge impact on the further development of number theory and algebra. The laws of reciprocity still occupy one of the central places in algebraic number theory. In Braunschweig, Gauss did not have the opportunity to get acquainted with the literature necessary for work on the Arithmetical Investigations. Therefore, he often traveled to nearby Helmstadt, where there was a good library. Here, in 1798, Gauss prepared a dissertation on the proof of the fundamental theorem of algebra. Gauss left behind four proofs of the fundamental theorem of algebra. He devoted his doctoral dissertation, published in 1799, to the first proof, entitled "A new proof of the theorem that any entire rational algebraic function of one invariable can be decomposed into real factors of the first and second degree." Gauss did not fail to pay attention to the gaps in Euler, and most importantly, he criticized the very formulation of the question, when the existence of the roots of the equations was assumed in advance. The first proof of Gauss, like that of d'Alembert, was analytic. In the second proof, performed by him in 1815, the famous mathematician again returned to the criticism of the proof of the fundamental theorem of algebra by means of reasoning, when the existence of the roots of the equation is assumed in advance. Gauss explained in the introductory paragraph the need for a new proof: "Although the proof of the factorization of an entire rational function, which I gave in a memoir published 16 years ago, leaves nothing to be desired in terms of rigor and simplicity, it is to be hoped that mathematicians will not consider it undesirable that I return again to this extremely important question and undertake the construction of a second no less rigorous proof, starting from completely different principles. on purely analytical principles. It should be noted that what Gauss calls the analytic method is today called the algebraic one. For the proof, Gauss used the construction of the expansion field of a polynomial. More than sixty years have passed when L Kronecker also improved and developed the Gauss method for constructing the expansion field of any polynomial. Subsequently, Gauss gave two more proofs of the fundamental theorem of algebra. The fourth and last refers to 1848. The main result of the proofs of the fundamental theorem of algebra by Euler, Lagrange and Gauss, I.G. Bashmakov, was that "algebraic proofs of the fundamental theorem of algebra are valuable precisely because for their implementation new deep methods of algebra itself were developed and the forces of already created methods and techniques were tested." Author: Samin D.K.
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