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In 1801, astronomers should be able to observe in the night sky that a group of outstanding mathematical geniuses were about to shine and create the greatest century in the history of mathematics. Among that group of dazzling geniuses, nothing shines more brightly than Nils Henryk Niels Henrik Abel. "he left mathematicians something that would keep them busy for 500 years," Hermit said of Abel. "

Abel was born in Norway on August 5, 1802. Abel's mother is extremely beautiful (Abel inherited her face). Like several other first-rate mathematicians, Abel discovered his talent early on. At the age of 16, he thoroughly understood the great works of his predecessors, including Newton, Euler and Lagrange, through self-study. A few years later, when asked how he quickly entered the ranks of the first class, he replied: learn from the masters.

Today we know that many of the things that the masters thought they had proved before the 19th century were actually not really proved at all. This is especially true of Euler's research on infinite series and Lagrange on analysis. Abel was acutely aware of the flaws in the reasoning of his predecessors, and he was determined to make up for them. One of his masterpieces in this respect is the first proof of the general binomial theorem. Newton and Euler have explained some special cases of this theorem, but it is not easy to prove this theorem reliably in general cases.

Abel's first ambition in mathematics was to solve the general quintic equation problem. The role of the general quintic equation in algebra is similar to the key experiment that determines the fate of a scientific theory. At the beginning of middle school algebra, we learned primary or quadratic general equations, such as

Then I learned about cubic and quartic equations, such as

For the general equation of degree 1 to degree 4, we obtain the finite formula of the solution, that is, the unknown number x is represented by the known coefficient a _ ~ ~ b ~ ~ c ~ e. Such solutions are called algebraic solutions. In this definition of algebraic solution, an important condition is "finite", that is, the formula solution formed by the combination of the coefficients of the equation through a finite number of four operations and root signs. After successfully solving the first quartic algebraic equation, algebraists struggled for almost three centuries to work out the algebraic solution of the general quintic equation.

But no one can succeed. This is where Abel starts. The following is an excerpt from Abel's paper on Algebraic Solutions of equations.

One of the most interesting problems in algebra is to find the algebraic solution of the equation. Therefore, almost all outstanding mathematicians have discussed this problem. We can easily get the general expression of the root of the first quartic equation. It is believed that a unified method of understanding the first quartic equation can also be applied to any equation, but despite all the efforts of Lagrange and other outstanding mathematicians, the expected goal has not been achieved. This leads to the conjecture that the general equation cannot be solved algebraically, but it is impossible to determine, because the exact conclusion can be drawn only if the equation is solvable.

……

The real scientific method to be adopted is rarely used because of the extremely complex calculations it requires, he continued:

I have explored several branches of analysis in this way, and although I often ask myself more questions than I can do, I have come up with a large number of general results that strongly illustrate the nature of quantities, and elucidating the nature of these quantities is the purpose of mathematics. In another paper, I will discuss the algebraic solution of the most general equations.

He discussed two related general issues:

Find out all algebraic solvable equations of any given number of times.

Determine whether a known equation is algebraically solvable.

Before Abel could return to these questions, his uncontrollable creativity urged him to consider a wider range of questions, and the complete answers to these questions were left to Galois to answer. Although Abel's work on algebra was epoch-making, it was overshadowed by his creation of a new branch of analysis. As Legendre said, this work is Abel's "everlasting monument".

In the 19th century, France was the mathematical queen of the world, while Germany had Prince Gauss of mathematics. In August 1825, 23-year-old Abel began a study trip to France and Germany with government funding. He brought his own paper, in which he proved that it was impossible to solve the general quintic equation by algebraic method. Abel naively believed that this was his science passport to mathematicians on the continent, and he particularly hoped that Gauss would see the importance of this work. But he did not know that the "prince of mathematics" sometimes did not show the magnanimity of a prince to young mathematicians who tried to be recognized.

Gauss received the paper as scheduled. But he threw it aside without a glance. From then on, Abel hated Gauss and denigrated him whenever he had a chance. He said that what Gauss wrote was obscure. Gauss is often criticized for his "arrogant contempt" in this matter. At that time, the problem of the general quintic equation was already well known. If today a mathematician receives the so-called solution of turning a circle into a square, he will not necessarily talk to the author. Because he knew that Linderman proved in 1882 that it was impossible to square a circle with a ruler alone. He also knew that Linderman's proof was acceptable to anyone.

In Abel's manuscript in 1824, the problem of the general quintic equation was almost the same as the problem of turning a circle into a square, so Gauss became impatient. But the problem has not yet been proved unsolvable, as Abel's article provides proof. If Gauss had been patient, he could have read something that interested him. As long as his words, Abel may become famous, and even Abel's life may be extended.

After leaving home in September 1825, Abel first visited famous mathematicians and astronomers in Norway and Denmark. Then I went to Berlin. In Berlin, he met a man named Auguste Leopold Krell, who would be his Bole in science. "Claire" has become the proper name for the magazine he founded, the first three volumes of which include 22 papers by Abel. Abel's great work made the magazine famous throughout the mathematical world; at last it made Claire famous.

Claire Claire himself is a math enthusiast, not a creative mathematician. He is a construction engineer by profession. He got ahead early in his work and built Germany's first railway. In his spare time, he studies mathematics carefully, not just as a hobby. The Journal of Pure and Applied Mathematics, which he founded in 1826, gave a great boost to German mathematics. The establishment of this magazine is Krell's greatest contribution to the development of mathematics. This magazine is the first regular publication in the world devoted to the results of mathematical research. From 1826 to the present, Claire has been published every three months with new mathematical articles. Today, hundreds of magazines are involved in all or a considerable part of the development of pure and applied mathematics.

Abel mentioned to Krell his proof that it was impossible to solve the general quintic equation by algebraic method. Claire wouldn't even listen; there must be something wrong with any such proof. But he accepted the paper, admitting that Abel's reasoning was above him-and finally published Abel's detailed proof in his Magazine.

The rich social activities in Berlin began to distract Abel from his work, and he hid in Freiburg, where he could concentrate on his work. It was in Freiburg that he forged his greatest work and created what is now called the Abel theorem. But he had to go to Paris to meet with Legendre, Cauchy and others, the first-rate French mathematicians of the time.

From a letter Abel wrote to astronomer Hanstin, it can be seen that Abel wants to reconstruct mathematical analysis:

In advanced analysis, few propositions have been proved extremely strictly. It is really interesting to find out the cause of this situation. In my opinion, the reason lies in the fact that most of the functions that have appeared in analysis so far can be expressed as power functions. When we adopt some general method, it is not too difficult; but I must be very careful, because propositions without strict proof have taken root in our minds, so that we often run the risk of adopting them without further examination. These trivial things will appear in Mr. Krell's magazine.

In October 1826, Abel's "on the General Properties of a very extensive Class of Transcendental functions" was submitted to the Paris Academy of Sciences. This is what Legendre calls the "Eternal Monument" and what Hermit calls "something that will keep future mathematicians busy for 500 years." It is a culminating achievement in modern mathematics.

Legendre and Cauchy were appointed reviewers of the paper. Legendre complained: "We found this paper difficult to read; it was written in almost white ink and the handwriting was so bad that we both thought we should ask the author to send a neatly readable copy." Cauchy took the paper home and put it somewhere and forgot all about it.

Jacobi exclaimed, "what a discovery of Mr. Abel!"... has anyone seen the same thing? this discovery, perhaps the greatest discovery of our century, was handed over to your Academy of Sciences two years ago, but how could your colleagues not notice it? " Cauchy turned it out in 1830. Finally, it was published in the Proceedings of famous scientists of the French Academy of Sciences, but it was already 1841.

The following are the first few paragraphs of the paper

So far, mathematicians have studied a very small number of transcendental functions. In fact, the whole theory of transcendental function is simplified to logarithmic function, trigonometric function and exponential function, which is essentially a simple class of functions. It is only recently that I began to consider some other functions. Among the functions considered later, elliptical transcendental functions occupy the first place, and their remarkable and exquisite properties are developed by Mr. Legendre. In his paper, Abel studied a wide range of functions, that is, all those functions whose derivatives can be represented by algebraic equations whose coefficients are univariate rational functions. He has proved some properties of these functions similar to logarithmic functions and elliptic functions. And the following theorem is obtained:

If we have several functions, their derivatives can be the root of the same algebraic equation, and all the coefficients of the equation are rational functions of a single variable, as long as we establish a certain number of algebraic relations between the variables of the functions discussed. We can always use an algebraic function and a logarithmic function to represent the sum of any number of such functions. The number of these relationships does not depend on the number of functions at all, but only on the properties of the specific functions being considered.

The theorem described so simply by Abel is now commonly known as the Abel theorem. Legendre didn't talk about elliptic functions. Legendre spent most of his life studying elliptic integrals. The difference between elliptic integrals and elliptic functions is like the difference between a horse and its cart, which is precisely the root of one of Abel's greatest contributions to mathematics.

Abel was the first to consciously reverse the problem. If the answer to a question is in a hopeless situation, try to turn the question upside down, using the question as the argument and the argument as the question.

In integral calculus, the inverse trigonometric function is naturally presented in the form of a simple definite integral. Such integrals occur when we try to find the arc length of a circle by using integrals. Assuming that inverse trigonometric functions appear in this way in the first place, wouldn't it be "more natural" to consider the inverse functions of these functions as known functions to be studied and analyzed?

But in many more advanced problems, the simplest problem is to calculate the arc length of an ellipse by integral, and the first thing that appears is the thorny inverse "ellipse" function. This made Abel see that these functions should be studied "the other way around". However, Legendre, a great mathematician, spent more than 40 years on his "elliptic integral" without once doubting that he should consider it the other way around. This extremely simple and ordinary way of looking at seemingly simple but actually esoteric problems is one of the greatest mathematical advances of the 19th century.

Abel found that the new functions derived from the inverse functions of elliptic integrals have two periods, and their ratios are imaginary numbers. After that, Jacobi, Rosenhein, Wellstrass, Riemann, and many others delved deeply into Abel's great theorem. By developing and expanding Abel's thought, they found some functions of n variables with 2n cycles. Abel himself explored his findings more deeply. His successors applied the whole work to some parts of geometry, mechanics, mathematical physics and other branches of mathematics, and solved some important problems that could not have been solved without the research that Abel began.

In the early morning of April 6, 1829, Abel died and lived only 26 years and 8 months. Two days after his death, Abel was appointed professor of mathematics at the University of Berlin.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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