At the beginning of the nineteenth century, the european mathematics community had been searching for it for hundreds of years, but had yet to find five equations. Garowa's paper proved to have dashed the hopes of the academic world, but it also opened the curtain of modern mathematics。
In 1832, knowing that galois was dying, he wrote a paper for almost half a century, with 32 pages of paper, and from time to time, “i don't have time”。
The next day, he died in a fight, and a weak, passionate genius left. He was 21
After 14 years, no one has been able to figure out exactly what galois wrote. It includes the best mathematicians of that time, physicists -- gaussy, cosy, fourier, lagrands, yabei, plumson, none of them really understood galois's theory。
No one thought that the 21-year-old boy's superstitious theory had created a river of modern-day mathematics。
“i'm sure it's the task of future mathematics to jump out of calculations, cluster calculations, classified according to their complexity rather than their appearance.” this sentence, left behind by garois, continues to pass through the night sky like lightning。
Group theory: the advent of modern generation mathematics
Why is mathematicians so obsessed with five equations? Because in five equations, mathematicians for the first time carved out modern science hidden under the iceberg and brought mathematics into a brilliant modern myth。
Communism has opened up a whole new battleground, replacing computation with structural research, moving away from thinking that focuses on computational research to thinking that looks at structure, and categorizing mathematical calculations, making it rapidly a brand-new mathematical branch, with a huge impact on the formation and development of recent generations。
And the emergence of a myth also laid the physical foundations of the 20th century. Since then, the newton mechanical cosmology, which ruled humankind for almost 200 years, has begun to move into random and uncertain quantum worlds and vast temporal relativism。
A scientific revolution of unprecedented magnitude has been stormy and has even continued to this day. Today's physics and mathematics are clearly out of the question of a day without homogeneity, and the intersection of arithmetic and tact is an extremely mysterious phenomenon in modern mathematics, in which the galois are playing an important role。
Careful observation of the garowa herd and the basics of the prowl reveals that they are very similar. In order to understand more deeply the nature of the scale, the top 20th-century math genius grottendik has developed the motive theory, which is still mysterious today, and garowa's theory can be seen here as a zero-dimensional special case。
Another view from a different angle is that the garowa (basic group) completely determines a particular type of geometric object, an anabelian theory advanced by grottendik。
And in algebra, the galois are the core object, and its integration with the expression theory is another grand building of modern mathematics -- – the dreams of the langetz platform, which are also organically linked to the motive theory mentioned above, together constitute what we call the vast blueprint of the programme in the geometry of arithmetic。
Five equations: is there a root formula
The historical dilemma that brought us back to the source of the cluster theory: is there a common solution to the five equations
This is essentially about the oldest and most natural problem in the history of mathematics: the root of the equation of one dollar and more。
Back in birun, cuba, people were able to solve the equation. Any secondary equation, now we'll use its root formula to solve it. And three equations and four equations were solved only in the middle of the 16th century, spanning more than three thousand years, and finally three equations -- the caldano formula -- were born in the middle of the battle between mathematicians such as tartalia, cardano and ferrari. The solution to the equations four times is much faster than expected, and ferrari has learned with great wisdom the three equations of master caldano, and has managed to obtain the root solutions of the equations four times by lowering them. In response, mathematicians have grown ambitious and are beginning to believe that all one dollar and many more equations can find the corresponding solution。
However, when everyone thought that the solution of the five equations would follow, it was silent for over 200 years。
The first one to provide new ideas for the solution of the five equations was the "cyclops" eurasia in mathematics, who converted the five equations of any full coefficient into a form. Ora thought it was possible to find five equations, but eventually nothing。
At the same time, mathematics genius lagrand is searching for five equations. Drawing on ferrari's historical experience of reducing the equation four times to three, he was made. Unfortunately, the same transformation has resulted in five equations for six。
Since then, mathematicians ' footsteps have been blocked by five equations, and progress in finding multi-dollar equations has once become a mystery. On the other hand, there were numerous arguments about the equation, which were focused on two main issues。
(1) is there at least one solution to the secondary equation
(2) if there is a solution to the next equation, how many will there be
Mathematical prince goss is out. In 1799, he proved that every equation had one solution. So he deduces that five equations necessarily have five solutions, but can they be expressed through formulas
With the fog out, the challenge is still before us, and the question of whether or not five equations are a solution continues to haunt humanity。
One wave and three times: the genius of dust
In 1824, abel published a paper entitled " five equations without algebras " , which for the first time gave an unsolvable proof of the normal five equations in their root form, which was the first time that humanity had actually encountered five equations to solve. Mathematicians are shaking their heads in the face of this unknown boy from northern europe, and they don't believe that the problem will be solved. After receiving the paper, cosy threw it in one of the drawers of the desk at random, leaving only one comment: “what kind of monster is this?”。
Despite the fact that this rare genius eventually died of illness, his paper succeeded in revealing the differences between the high and low equations, proving that the five-algebra equations did not exist. This proof by abel has freed mathematics from the ideological constraints of equations seeking to solve and rooting, and has been subversive in suggesting that a formula that is expressed uniformly through the additions and subtractions and openings of equations cannot be used to solve the normal five equations。
How can equations be distinguished and determined that they can be solved in a simple algebraic formula (the coefficient root formula) and which equations cannot? Abel did not give the perfect answer to this question. It was not until galois was born that the solution to the high equation really fell。
In 1830, 19-year-old garowa opened a wider world of abstract algebras with a paper. He introduced a new concept — the cluster — in a more complete and powerful way, which proved that a sufficient requirement for a one-dimensional equation to solve by root is that the galois group of the equation be soluble (limited)。
Because the normal one-dollar-altitude galois is a symmetrical group of words, and when it is not a solventable group, this is the underlying reason why four equations can be solved, while five equations are so high (more than four times)。
Garowa opened the realm of “group theory” hidden for centuries, and he gave his paper to the mathematician at the time, cosy, whose treatment was no different from that of abel, who promised to forget it at the end of the day, and even lost galois's abstract。
Garois wrote the equation in three articles, presenting the material with confidence for the grand prize in mathematics, but when it was sent to fourier, fourier died, and garois's paper was once again dusted。
With the encouragement of parsons, galois presented new papers to the french academy of sciences, while the two-faced persons said that the theory of galois was “ununderstood”. The youthful and talented galois anger felt that there was no point in the field of mathematics, putting all its power into the political movement and saying "if a body is needed to awaken the people, i will give myself up." the best genius in the field of mathematics has become the wrath of a new era, full of anger against the world。
Then, by coincidence, galois met the goddess of his life in political activities and turned his mind upside down and went into flames. This is a mysterious woman with a married woman, whose husband is of the same character as garois, who is angry and fighting over it. In the end, galois died in a duel。
Perhaps the gods felt guilty about garova's fate, and in the middle of the day gave garova the opportunity to prepare his last words before he died and entrusted the results to his friend august chevalier. Friends did not answer the question and sent their manuscripts to goss and yabei without receiving a response. In 1843, french mathematician liu wei wei wei had a vision that not only confirmed galois’s philosophy, but also made public the root causes of five unsolved equations. So, the talent and contributions of galois are known。
Galois: a new chapter in mathematics
In fact, abel and garowa did not prove that five equations were unsolved, but rather that a more subtle thing was to assume that they existed, but none of the algebra operations (plus or minus any of the subsides) were sufficient to express them. In retrospect, the solution of the low equation mentioned above can be expressed only in algebra operations. In doing so, gallowa demonstrated his amazing talent and keenly sensed that the symmetry of multiform solvency could be observed by the multiplicity itself without having to solve it, and that symmetry itself determined entirely whether or not it existed。
By observing these formulas, garova notes that the organization of these roots in an arbitrary manner, such as in turn and in a converse manner, does not change this expression, and that the various elements are arranged in different ways, but the sum remains unchanged. There are 120 different forms of ranking of the five numbers, so there are 120 symmetrys of a standard five. In order to describe this symmetry, galois created the concept of a cluster. According to a constellation of 120 arrays, which does not permit the appearance of the towers called for by the equation, garowa has demonstrated that a five-formula equation with root resolution is allowed to top 20。
In this way, garowa actually solved the problem that abel had not resolved and provided clear criteria for determining which equations had been solved and which did not. If there's a polygon in front of you now, it's got no more than 20 elements in the galois herd, then it's root solvency。
With the discovery of garowa's five equations, garowa continues to be a thorn, successfully proving that it is incomprehensible when the interlocking of the swarms is a single one that cannot be exchanged. The general submersible galois is a subset of subsymmetrical clusters, so it is not possible to decipher the equations by root in five or more terms。
If there's still some ambiguity here, we'll read it in detail. The setting is the field of an unorthodox, assuming it is divided. As a dividing field, the galois are actually the replacements on the root cluster, which correspond to some of the necessary intermediate equations of the equation. The equation is sufficiently necessary for root solvency to be soluble for the galois. So when time is incomprehensible。
The galloway methodology, which was demonstrated by the use of herds, was successful in breaking the mystery of the solvency of the equation, and clearly illustrated why there was no root solution for the equations above four, while the equations of four and below had its roots decomposition, which even completed a longitudinal journey and solved two of the three ancient mapping problems, namely, “no three-point arbitrary angles” and “no double cubes”。
These have made a great contribution to the mathematics community, and the introduction of concepts such as the “cluster” “domain” has been the birth of abstract algebras. As a result, the results of galois have been codified into garois theory. The galoisian theory, which has evolved to the present, has not met expectations and has become one of the fundamental pillars of contemporary algebra and digitism, with outstanding merit。
French mathematician bica, commenting on the achievements of mathematics in the nineteenth century, said, “no one can compare to the originality and depth of garois' concepts and ideas.” looking back at the solution of the five equations, the symmetrical, localismal and convoluted, it is no wonder that the top review masters of the academic world were in the fog。
This theory of sweat and life finally bloomed after more than 300 years of success in the three equations that had plagued humanity for centuries, and was eventually answered by the galois theory。
