16th century has successfully solved the problem of solving equations with roots of quartic and below, but in the following centuries, the general solution of equations with quintic and above has not been obtained. Around 1770, the French mathematician Lagrange reformed the thinking method of algebra, and put forward that the replacement theory of equation roots is the key to solving algebraic equations. By using Lagrange's resolvent method, that is, using any n-degree unit root of 1, the resolvent formula x 1+X2+2 x3+…+N- was derived. His work strongly promoted the development of algebraic equation theory. But his method can't give a radical solution to the general quintic equation, so he suspects that there is no radical solution to the quintic equation. Moreover, he also failed in finding the algebraic solution of the general equation of degree n, thus realizing that the general algebraic equation of degree 4 or more cannot have a radical solution. His thinking method and the method of studying root substitution give inspiration to future generations.
In 1799, Rufini proved that the resolvent of an equation with more than five degrees cannot be less than four, thus proving that the equation with more than five degrees cannot be solved by roots, but his proof is not perfect. In the same year, German mathematician Gauss opened up a new method. In proving the basic theory of algebra, he did not calculate a root, but proved its existence. Later, he began to discuss the concrete solution of higher-order equations. In 180 1, he solved the cyclotomic equation xp- 1=0(p is a prime number), which shows that not all higher-order equations can be solved by roots. Therefore, whether all higher-order equations or some higher-order equations can be solved by roots needs to be further clarified.
Subsequently, the Norwegian mathematician Abel began to solve this problem. From 1824 to 1826, Abel began to study the properties of roots of equations that can be solved by roots, so he corrected the defects in Ruffini's proof and strictly proved that if an equation can be solved by roots, then every root in the expression of roots can be expressed as the root of the equation and the rational number of some unit roots. Abel theorem is proved by this theorem: in general, equations higher than quartic cannot be solved by algebraic method. Then he further thought about which special higher-order equations can be solved by roots. On the basis of the solvability theory of Gauss cyclotomic equation, he solved the solvability problem of an arbitrary special equation, and found that all the roots of this special equation are rational functions of one of the roots (assuming X), and any two roots q 1(x) and q2(x) satisfy q 1q2 (X) = Q2Q6544. Now this kind of equation is called Abel equation. In fact, some ideas and special results of groups have been involved in the study of Abel equation, but Abel failed to recognize and clearly construct the permutation set of the roots of the equation (because if all the roots of the equation are expressed as rational functions QJ (x 1), J = 1, 2, 3 ..., n, when the other one is used, it should actually be said that the root Xi = Q. Q2(xi), …, qn(xi) is an arrangement of roots x 1, x2, …, xn), and only the interchangeability q 1q2(x)=q2q 1(x
Abel solved the problem of constructing algebraic solvable equations of any number, but failed to solve the problem of judging whether known equations can be solved by roots. It is against this background that the French mathematician Galois began to take over Abel's undisputed career.
Galois, like Lagrange, began with the replacement of the roots of the equation when he proved that there was no general root solution for the equation of degree five or higher. When he systematically studied the permutation and substitution properties of the roots of equations, he put forward some criteria to determine whether the solutions of known equations can be obtained from the roots. However, these methods only made him consider an abstract algebraic theory called "group". In the paper 183 1, Galois first put forward the term "group", called the set of closed permutations a group, and defined the concept of permutation group for the first time. He believes that understanding permutation groups is the key to solving the equation theory, and the equation is a system, and its symmetry can be described by the properties of groups. From then on, he began to solve the problem of equation theory and study group theory directly. He introduced many new concepts about group theory and produced his own Galois group theory, so later people called him the founder of group theory.
For rational coefficients, the equation x+axn-1+a2xn-2+…+an-1x+an = 0 (1).
Suppose every transformation of its n roots x 1, x2, …, xn is called a permutation, and its n roots * * * have n! There are five possible permutations, which are multiplied by the set of permutations to form a group, which is the permutation group of roots. The solvability of the equation can be reflected in some properties of permutation groups of roots, so Galois transformed the solvability of algebraic equations into the analysis of the properties of permutation groups and their subgroups. Now the permutation group associated with the equation (indicating the symmetry of the equation) is called Galois group, which is a group in the coefficient field of the equation. For every polynomial function with a rational function value about the root, the Galois group of the equation is the largest permutation group that meets this requirement. In other words, any polynomial function about the root has a rational value, and every permutation in Galois Group keeps the value of this function unchanged. Galois founded group theory to be applied to equation theory, but he was not limited to this, but extended the group theory and applied it to other research fields. It's a pity that Galois's group theory is too abstruse for people at the beginning of the 9th century to understand/kloc-0. Even mathematicians at that time could not understand the essence of his mathematical thought and work, so that his paper could not be published. More unfortunately, Galois 2 1 died young because of a stupid duel. We have to feel sorry for this genius. It was not until the 1960s that his theory was finally understood and accepted by people.
Galois group theory is recognized as one of the most outstanding mathematical achievements in19th century. He provided a comprehensive and thorough answer to the solvability of equations and solved the problem that puzzled mathematicians for hundreds of years. Galois group theory also gives a general method to judge whether geometric figures can be drawn with straightedge and compass, which satisfactorily solves the problem that bisecting any angle or cube is insoluble. Most importantly, group theory has opened up a brand-new research field, replaced calculation with structural research, changed the way of thinking from emphasizing calculation research to using structural concept research, and classified mathematical operations, which made group theory develop rapidly into a brand-new branch of mathematics and had a great influence on the formation and development of modern algebra. At the same time, this theory has a great influence on the development of physics and chemistry, and even on the emergence and development of structuralist philosophy in the twentieth century. http://www.nhyz.org/psz/%CA%FD%D 1%A7%CA%B7/buer.html