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Hermit was born in France on December 24, 1822. Creative talent is rarely combined with the ability to master the quintessence of other people's theories. What was needed in the middle of the 19th century was to coordinate Gauss's arithmetic creation with the discoveries of Abel and Jacobi in elliptic functions and the theory of algebraic invariants developed by the British mathematicians Bull, Kelley and Sylvester. It is this ability of Hermit.

When Hermit was a middle school student, he mastered Lagrange's paper on solving digital equations in the library. He also bought Gauss's "arithmetic research" and mastered it, but before or after that, only a few people mastered it. After Hermit understood the mathematical research that Gauss had done, he was ready to start his own research. He later said that I learned algebra from the works of Lagrange and Gauss.

The first volume of the New Mathematical Annals (founded in 1842) contains two articles that Hermit wrote when he was a student. The first article is about the analytic geometry of conic curve, which shows little originality. The second part is "discussion on the algebraic solution of quintic equation".

Hermit said

Lagrange makes the algebraic solution of the general quintic equation depend on determining the root of a special quartic equation, which he calls the simplified equation (today known as the resolvent equation). Therefore, if the simplified equation can be decomposed into quadratic or cubic rational factors, we will get the solution of the quintic equation. I will try to show that such a decomposition is impossible.

Hermit succeeded in his attempt and joined the ranks of algebraists. It is strange that Hermit thinks that elementary mathematics is difficult. His grades at school are mediocre.

In the second half of 1842, Hermit took the entrance examination of the Comprehensive Engineering School at the age of 20. But the result only ranked 68th. The exam became a "stain" on the young maths master, and all his future successes could not be eliminated.

Hermit spent only one year at the comprehensive engineering school, and that year he spent all his time on the Abel function. Abel function was the focus and focus of mathematics research at that time. He also met a first-rate mathematician, Liouville.

Hermit's pioneering work on Abelian functions began before he was 21. In 1843, Hermit wrote a letter to Jacobi

Studying your paper on quadruple periodic functions generated from the theory of Abelian functions, I have come to a theorem about the separation of variables of these functions, similar to what you have given. The theorem of the simplest expression of the root of the equation discussed by Abel is obtained.

Let me give a general explanation of the nature of the problem. A trigonometric function is a function with a variable and a period.

Where x is a variable and 2 π is a period; by "reversing" the elliptic integral, Abel and Jacobi found a function with one variable and two periods, such as

Where p and Q are periods; Jacobi found two variables and four periodic functions, such as

Where a _ r _ r _ b _ c _ r _ d is a cycle. One of the problems encountered in the early days of trigonometry was expressed in sinx.

Where n is an arbitrary integer. What Hermit wants to solve is the corresponding problem of a function with two variables and four periods; in Hermit's unparalleled more difficult problem, the result is still an equation, about which, surprisingly, it can be solved algebraically, that is to say, with a root.

Hermit not only shared his findings on Abelian functions with Jacobi, but also wrote him four long letters about number theory. The first of these letters, written when Hermit was only 24, opened up a new field (we will point out in a moment), and these letters alone are enough to establish Hermit as a creative first-rate mathematician.

Jacobi proved that the following assertion that ∶ has three different periods of univariate single-valued functions is impossible. A single-variable single-valued function with one or two periods can exist.

A single-valued function takes only one value for each value of a variable.

Hermit claimed that Jacobi's theorem introduced him to the idea of advanced arithmetic. These methods are too professional to describe here, but you can simply point out the ideas.

The arithmetic in the sense of Gauss, and discusses the properties of rational integers. In particular, Gauss studied the integer solutions of indefinite equations with two or three unknowns, such as in

In, a _. It should be noted here that the problem is definite and has to be solved entirely in the rational integer field. It seems impossible to study such a discrete problem if it is used to study the "analysis" of continuous numbers, which is exactly what Hermit does. He started with the discrete system representation, applied the analysis to this problem, and finally got the results in the discrete field. Since analysis is more fully developed than any discrete method, Hermit's work is comparable to the introduction of modern machines for medieval handicrafts.

In terms of algebra and analysis, the methods used by Hermit are much more powerful than those that Gauss could use when writing arithmetic Research. These more modern methods enabled Hermit to solve the problems that puzzled Gauss in 1800. In one development, Hermit caught up with the type of general problem discussed by Gauss and Eisenstein, and he began at least the arithmetic study of the quadratic form of any number of unknowns.

The general nature of arithmetic "type theory" can be seen in the statement of a particular problem. The Quadratic Gauss equation instead of two unknown quantities (x ~ (1) y)

An integer solution of a similar equation requiring s unknown quantities of degree n, where nline s is an arbitrary integer and every term on the left side of the equation is of degree n. Hermit describes how, after careful consideration, he saw that Jacobi's periodic study of single-valued functions depended on some more profound problems in the theory of quadratic forms.

In this infinitely vast field of research that Mr. Gauss shows us, algebra and number theory seem bound to be integrated into the same order of analytical concepts, and our current knowledge is not enough for us to form an accurate idea about it.

For x ^ 3-1x 0, understand

It is both sufficient and necessary; for x ^ 7 + ax+b=0, where aforme b is any known number, what kind of "number" x must be invented in order for x to be clearly expressed in aformab? Gauss provides a class of solutions ∶ any root x is a complex number. But this is just the beginning. Abel proved that if only a finite number of rational operations and squares are allowed, then there is no explicit representation of x as a meme b. We will return to this question later; Hermit made one of his greatest discoveries somewhere in his mind even earlier.

An arithmetic study by Hermit (though it is quite professional) can be mentioned here as an example of the prophecy of pure mathematics. We recall that in order to give the simplest expression to biquadratic reciprocity, Gauss introduced complex integers into higher arithmetic. Then Dirichlet discussed some quadratic forms in which rational integers, which appear as variables and coefficients, are replaced by Gaussian complex integers. Hermit explored the general situation of this situation and studied the representation of integers in what is called the Hermitian form today. An example of such a form (a special case in which n variables are replaced by two complex variables xdistinct 1 and their conjugations) is

Axiom 12 and axiom 21 are conjugate, and axiom 11 and axiom 22 are their respective conjugacy (so axiom 11 and axiom 22 are real numbers). It is very easy to see that if you multiply all the products, the whole form is real (without I).

When Hermit invented such forms, he was interested in discovering what numbers were represented by these forms. More than 70 years later, it has been found that Hermitian algebra is indispensable in mathematical physics, especially in modern quantum theory. Hermit did not know that his pure number society became valuable in science long after his death.

Hermit's invention in the theory of algebraic invariants is too professional to be discussed here. We introduce his two amazing achievements in other fields. Hermit found some of the most surprising originality of all his work in two areas: general quintic equations and transcendental numbers. The properties he found in the first field are clearly reflected in the introduction to his essay on the solution of the General Quintic equation:

As we all know, the general quintic equation can be reduced from the coefficient to the following form without any irrational substitution except the square root and the cubic root.

That is to say, if we can solve this equation, then we can solve the general quintic equation.

This remarkable result, thanks to the English mathematician Jerrard, is the most important step in the algebraic theory of quintic equations since Abel proved that root solutions are impossible. The impossibility proved by Abel indicates that it is necessary to introduce some new analytical elements (some new function) when looking for a solution, so it seems natural to use the root of the very simple equation we have just mentioned as an auxiliary quantity. However, in order to prove that it is reasonable to use it strictly as a basic element in the solution of a general equation, it is also necessary to know whether this simplicity of form can lead us to some ideas about the properties of its roots. We know nothing about these quantities except the fact that they cannot be expressed in roots.

It is now worth noting that Gerald's equation is greatly and easily applied to this study, and there may be a real analytical solution in the sense that we will explain it. Because we may indeed consider the algebraic solution of the equation from a point of view different from that shown by the solution of the first quartic equation for a long time.

Instead of using a formula including multi-valued roots to represent closely related roots that are considered to be functions of coefficients, we can try to obtain roots represented by single-valued functions of several different auxiliary variables, the number of these variables is as many as that used in cubic equations. In this case, the equation under discussion is

As we know, as long as the coefficient an is represented by the sine of an angle (for example, A), it is sufficient to express the root of the equation as the following determined functions.

Hermit reviewed the "trigonometric solution" of the cubic equation here. The "auxiliary variable" is A; the "single-valued function" is the sine function here.

Now the relevant equation is

What we have to show is a completely similar fact. It's just that you have to use elliptic functions instead of sine and cosine functions.

Then Hermit immediately began to solve the general quintic equation, using elliptic functions for this purpose. It is almost impossible to explain the problem to non-mathematicians. To use a very inappropriate analogy, Hermit discovered the famous "lost chord", and no one had the slightest suspicion that such an unpredictable thing would exist somewhere in time and space. His completely unexpected success caused a sensation in the field of mathematics. What is more remarkable is that it has created a new department of algebra and analysis, in which the main problem is the discovery and study of functions according to which general equations of degree n can be solved explicitly in finite form. The best result so far was obtained by Poincare, a student of Hermit, who created the functions that provided the solution needed. These functions are actually "natural" generalizations of elliptic functions. The generalized functions are characterized by periodicity.

Another sensational achievement of Hermit in the field of mathematics is to prove the transcendence of the base e of the natural logarithm.

E is about 2.718281828. E appears everywhere in today's mathematics (pure mathematics and applied mathematics). The concept of "transcendence" is extremely simple and important. Any root of an algebraic equation whose coefficient is a rational integer is called an algebraic number. A number that is not an algebraic number is called a transcendental number.

Now, given any "number" constructed according to some definite law, it is a meaningful question to ask whether it is algebraic or transcendental. For example, consider the following simply defined number

Among them, the index is 2pm, 6pm, 24pm, 120, … It is a successive "factorial", that is, 2 × 1 × 2, 6 × 2 × 3, 24 × 1 × 2 × 3 × 4120, 1 × 2 × 3 × 4 × 5,. Is this number the root of any rational integral algebraic equation? But the answer is no. On the other hand, by infinite series

The definite number is an algebraic number; it is the root of 99900x-1=0.

The first person to prove that some numbers are transcendental numbers is Liouville, who discovered a wide range of transcendental numbers in 1844, in which all forms are

Those numbers are the simplest transcendental numbers. But it is very difficult to prove that a particular number, such as e or π, is transcendental or not. So when Hermit proved that e is a transcendental number in 1873, the mathematical community was not only delighted, but also surprised by the incredible ingenuity of the proof.

Since then, it has been proved that most of them are transcendental numbers. In 1934, the young Russian mathematician Alexis Gelfond (Alexis Gelfond) proved that all types are

Is a transcendental number, where an is not 0 and 1p is an arbitrary irrational algebraic number. This solved the seventh question of Hilbert's question 23.

After Hermit proved that e is a transcendental number, Linderman of the University of Munich proved that π is a transcendental number, and his method is very similar to Hermit's method of proving e. In this way, the problem of "turning a circle into a square" will be solved forever. It is inferred from Linderman's proof that it is impossible to draw a square with an area equal to any given circle only with a ruler. This problem has been afflicting generations of mathematicians since Euclid's time.

Attachment: proof of the transcendence of e (Hilbert version)

First, suppose e is an algebraic number of degree n:

Equation 1: if e is an algebraic number of degree n, it satisfies this equation.

We use rational numbers to approximate the power of e and define the following objects:

Equation 2

Among them

For each power of e in equation 2, there are:

For the very small partial function, this equation means that all e ^ t is very close to a rational number. Now we replace equation 2 into equation 1 and eliminate the factor M to get:

Equation 3

Notice that equation 1 and equation 3 are the same quantity. Equation 3 has two obvious characteristics:

The expression in the first parenthesis is an integer and M makes the expression non-zero

In the second expression, benchmark will be selected small enough that the absolute value of the expression is < 1

Formula 1

Define M and floor

Hermitt first defined M and Austria. First, he defines M as:

Where p is a prime number. The prime number p can be as large as we want (but M is an integer for any value of p). The other definitions of M and floor are:

Equation 4

We will now continue to select p to satisfy the properties 1 and 2 above.

Let's first find the value of the integral M, multiply the binomial on the molecule, and get it.

It has an integral coefficient. Replace this with M, and then use

Get:

Limited to prime numbers greater than n, we will soon find that the first term of this equation is not divisible by p. However, we will soon find that the second item can. Expand the factorial:

Because M is not divisible by p, the first parenthesis in equation 3 is not divisible by p either. Now consider the first integral in equation 4. Introduce the variable y:

The integral becomes:

The polynomial in molecular parentheses has the integral coefficient term from

After several steps, we get:

For integer cs. Every M (k) is an integer divisible by p, so the first parenthesis in equation 3 is not divisible by p. Therefore, we conclude that the term in the first parenthesis of equation 3 is a non-zero integer. If it's 0, it's divisible by p, but we've come to the conclusion that it can't.

The last part of the rest is to prove that formula 1 is correct as long as we choose a large enough p value. Using equation 4, after several steps, we find that:

If the absolute value of this binomial product has an upper bound B for x ∈ [0jinn], we get:

Because p → infinite time RHS → 0, it is proved.

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

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