Shulou(Shulou.com)11/24 Report--

As a result, we can see that ideal prime factors reveal the nature of plural numbers, seem to make them easy to understand, and expose their internal transparent structure. -- Cuomer

Most of my readers will be disappointed to learn that as a result of this mundane observation, the continuous secret is about to be revealed. Dai Dejin

In the past 2000 years, few great mathematicians have made such efforts on the number theory of "pure number". There are many reasons. First, number theory is more difficult than other major fields of mathematics; second, the direct application of number theory to science is very little; third, mathematicians can make remarkable achievements in analysis, geometry and applied mathematics.

Modern arithmetic, after Gauss, began with the German Cuomer. Cuomer's theory began when he tried to prove Fermat's Great Theorem.

When Cuomer was 18, his mother sent him to Harley University to study theology. Because of poverty, Cuomer could not live in the university, but traveled back and forth between home and school every day. Heinrich Ferdinand Schake was a professor of mathematics at Harley at the time. Schake was obsessed with algebra and number theory, and he passed it on to Cuomer. In his junior year of college, Cuomer solved a difficult problem in mathematics and was awarded a doctorate at the age of 21.

Cuomer is one of the rarest scientific geniuses, his genius is embodied in abstract mathematics, applied mathematics and experimental physics. His most outstanding achievement is in number theory, where his profound originality prompted him to make some of the most important discoveries, while in other fields (analysis, geometry, applied physics), he also made outstanding contributions.

The basic theorem of arithmetic is that any natural number N greater than 1 can be uniquely decomposed into the product of finite prime numbers if N is not a prime number.

In the aspect of number theory, Cuomer reconstructed the basic theorem of arithmetic through the field of algebraic numbers (he completed this reconstruction through a new class of numbers, that is, what he called "ideal numbers"). He also continued Gauss's work on the biquadratic law of reciprocity and looked for laws of reciprocity higher than quartic.

The algebraic number field is produced by Kumer in proving Fermat's Great Theorem and Gauss Secant Circle Theory.

Cuomer's "ideal number" has now been replaced by Dedkin's "ideal number". By using his ideal number, Cuomer proved the equation.

For a wide class of primes p, there is no non-zero integer solution. He failed to prove Fermat's theorem. However, Cuomer took a big step forward in proving Fermat's Great Theorem, far more than all his predecessors had done.

Fermat's Great Theorem means that when the integer p > 2, the above equation has no positive integer solution. The p here is not necessarily a prime.

Kumer's last paper on Fermat's Great Theorem is the proof of Fermat's Theorem on the impossibility of an infinite number of primes p.

Cuomer is a bit like Gauss. He likes pure mathematics as much as applied mathematics. Cuomer developed Gauss's work on hypergeometric series and greatly developed Gauss's research, which is very useful in today's theory of differential equations.

In addition, Hamilton's wonderful work on ray groups (in optics) inspired Cuomer to make one of his own best discoveries, that is, the discovery of a quartic surface named after him. When the Euclidean space is four-dimensional (rather than the three-dimensional as we usually imagine), this surface plays an important role in the geometry of Euclidean space. Just like what happens when a straight line replaces a point as an irreducible element that makes up space. In 19th-century geometry, this surface (and its extension to high-dimensional space) occupies a central position, which can be represented by quadruple periodic functions. Jacobi and Hermit made the most important contributions to these functions.

Since 1934, Arthur Eddington has discovered that Cuomer's surface is related to the Dirac wave equation in quantum mechanics (both have the same finite group. Cuomer's surface is a wave surface in four-dimensional space). Cuomer returned to physics to complete this cycle because of his study of ray groups, and he made an important contribution to the theory of atmospheric refraction.

Cuomer spent the last nine years of his life in complete retirement. When he retired, he gave up math forever, except for the occasional trip to the place where he lived as a teenager. He died on May 14, 1893 at the age of 83 after a brief attack of the flu.

Dedkin Cuomer's successor in arithmetic is Julius William Richard Dedkin (Julius Wilhelm Richard Dedekind). Dedkin is one of the greatest mathematicians and most creative people in Germany. By the time Dedkin died in 1916, he had been a master of mathematics for far more than a generation. As Edmund Landau said,

Richard Dedkin is not only a great mathematician, but also one of the greatest mathematicians in the history of mathematics, the last hero of a great era and the last student of Gauss. From his works, not only we, but also our teachers, our teachers, have drawn inspiration.

Dedkin was born in Brunswick, where Gauss was born. By the age of 17, he had found many suspicious things in the so-called reasoning of physics and turned to mathematics, which is less controversial in logic. He entered Carolina College in 1848 (the same college that offered young Gauss the opportunity to study mathematics by himself). In this college, Dedkin mastered the principles of analytic geometry, advanced algebra, calculus and higher mechanics. He entered the University of Gottingen at the age of 19, and his main mentors were Moritz Abraham Stern, Gauss and physicist William Weber. From these three men, Dedkin received comprehensive and basic training in calculus, advanced arithmetic principles, least square method, advanced geodesy and experimental physics.

In 1852, Dedkin received his doctorate from Gauss for a short paper on Euler integrals (he was 21 at the time). Gauss' opinion on this thesis is very interesting, ∶.

This paper, written by Mr. Dade King, is about the study of integral calculus. It is by no means ordinary. The author not only shows that he has rich knowledge in the relevant fields, but also indicates the independence of his future achievements. As an examination article that has been approved for the examination, I think this paper is completely satisfactory.

In 1854 Dadkin was appointed an unpaid lecturer at the University of Gottingen, a position he held for four years. When Gauss died in 1855, Dirichlet moved from Berlin to Gottingen. During his last three years in Gottingen, Dedkin listened to Dirichlet's most important lecture. He also became a friend of the budding Riemann at that time. In 1857, Dedkin taught a course on Galois equation theory, which may be the first time Galois theory officially appeared in college courses.

Dade King was the first to attach importance to the concept of group in algebra and arithmetic. In this early work, Dade King has shown two main features of his later thought, namely abstraction and universality. He does not discuss groups on the basis of the permutation representation of finite groups, but uses axioms to define groups and tries to get their properties from the refinement of their essence.

Dedkin was appointed permanent professor at the Zurich Institute of Technology at the age of 26 for five years and returned to the Brunswick Institute of Technology in 1862, where he worked for half a century. Dade King worked in a relatively low position for 50 years, while some people who were not worthy to tie his shoelaces occupied important and influential university seats. Dade King remained unmarried until he died at the age of 85 (1916).

Dai Dejin's mathematical activities of dividing Dai Dejin are almost entirely related to the category of numbers in the broadest sense, and only his two greatest achievements can be discussed here. First of all, we describe his important contribution to the theory of irrational numbers and thus to the basis of analysis, that is, "Dedeking division".

Briefly review the properties of irrational numbers. If aforme b is an ordinary integer, the fraction a / b is called a rational number; if there is no integer m _ line n, such that a definite number N can be expressed as m / n, then N is an irrational number. If an irrational number is represented by decimal notation, then it is an infinite non-recurring decimal. The question is, how to use decimal counting to express irrational numbers and make them equal to real irrational numbers? Dedeking's definition of equality between numbers, rational numbers, or irrational numbers is consistent with that of Odox.

Odox, born in Nades about 400 BC, was a Greek astronomer and mathematician.

If two rational numbers are equal, then there is no doubt that their square roots are also equal. So, 2 × 3 and 6 are equal, so

But

The simple equation of this assumption

It's taken for granted in school arithmetic, and if we look at what this equation implies, it's obvious that ∶ calculates the square roots of 2, 3 and 6 (infinitely acyclic decimals), and then multiplies the first two together, and the result is equal to the third square root. Since these three roots cannot be accurately expressed no matter how many decimal places they are calculated, it is obvious that they can never be proved by the multiplication just described.

The continuous hard work of the whole of mankind in the whole process of its existence can never be proved in this way.

To make the concepts of "approximation" and "equality" precise to replace our initial rough concept of irrational numbers, which is what Dedkin did in the early 1970s. His work on continuity and irrational numbers was published in 1872.

The core of Dade King's irrational number theory is his concept of "segmentation" or "truncation" ∶.

A Dedkin partition divides rational numbers into two sets An and B, so that all elements of An are less than all elements of B, and A does not contain the largest element (open set). Set B may or may not have the smallest element in a rational number. If B has the smallest element, the partition corresponds to the rational number. Otherwise, this partition defines a unique irrational number. In other words, A contains all rational numbers that are less than partition, and B contains all rational numbers that are greater than or equal to partition. The partition is equal to an irrational number that does not exist in both sets. Each real number is equal to one and only one rational number.

In this way, each partition does define an irrational number. From the point of view of the true nature of irrational numbers, it is necessary to thoroughly understand mathematical infinity before establishing an irrational number theory. Infinite classes are obviously needed in Dejin's partition definition, and such classes will lead to serious logical difficulties.

Mathematicians have different views on whether these difficulties affect the consistent development of mathematics. Without a consistent mathematical infinite theory, there would be no irrational number theory; without irrational number theory, there would be no mathematical analysis of any kind that is even slightly similar to what we have now; and finally, if there is no analysis, then most of the mathematics that exists today, including geometry and most applied mathematics, will cease to exist.

Therefore, the most important task facing mathematicians seems to be to construct a satisfactory infinite theory. Cantor tried it and achieved great success.

Algebraic number

Another outstanding contribution made by Dade King to the concept of "number" is in the field of algebra. For the properties of the basic problems discussed, we must mention that algebraic number fields and algebraic integers are decomposed into prime factors. The crux of this problem is that the prime factors decomposed into in some of these fields are not as unique as in ordinary arithmetic. Dedkin restored the "uniqueness" he wanted by creating what he called an "ideal". An ideal is not a number, but an infinite class of numbers, so Dedekin returned to "infinity" to overcome his difficulties.

The concept of ideal is not difficult to understand, although it is contrary to common sense, but common sense will always be affected. An ideal must do at least two things ∶ it must actually let ordinary (rational) arithmetic take its course; it must force algebraic integers to obey the basic laws of arithmetic (uniquely decompose into primes).

The more inclusive classes divide the less contained classes, which involves the following phenomenon (and its generalization). Consider the fact that 2 is divisible by 4. Instead of the obvious fact (if you enter the algebraic number field, it gets nothing), we replace 2 with all classes of integer multiples of 2. For convenience, we use (2) to represent this class. Similarly, use (4) to represent all integer multiples of 4. It is now clear that (2) contains more classes; in fact (2) contains all the numbers in (4). The fact that (2) contains (4) is represented by symbols.

It is easy to see that if mline n is an arbitrary ordinary integer, then if and only if m divides n, there is

This reminds us that the concept of general arithmetic divisibility can be replaced by concepts contained in the class (just described), but it is useless if this substitution does not retain the unique properties of arithmetic divisibility. It can be explained in detail that it does maintain these properties, and here is only one example. If m divides n N divides l, then m divides l. For example, 12 divides, 24 divides, divides 24, divides 72, divides 72 exactly. Convert to the class as above, which becomes ∶ if the class (m) contains the class (n), and if the class (n) contains the class (l), then the class (m) contains the class (l). The result is that when we add the definition of "multiplication" ∶ (m) × (n) as a class (mn), such as (2) × (6) = (12), the numbers are replaced by their corresponding classes, and what is needed is done. Note that the above multiplication is defined and does not mean that it can be derived from the meaning of (m) and (n).

Dejin's ideal of algebra is a generalization of the above. Dedeking gives an abstract definition, a definition based on essential attributes, rather than based on a specific pattern that represents or describes what is defined.

Consider the set (or class) of all algebraic integers in a given algebraic number field. There are some subsets in this all-inclusive set. A subset is called an ideal if it has the following two properties.

The sum and difference of any two integers in the subset are still in the subset.

If any integer in the subset is multiplied by any integer in the all-inclusive set, the resulting integer is still in that subset.

Thus, an ideal is an infinite class of integers. It is easy to see that according to the ideal definition, the previously defined (m), (n), … They are all ideals. If an ideal contains another ideal, it is said that the first ideal divides the second.

It can be proved that every ideal is shaped as

A class of all integers, where axi1, … , a_n . It is a fixed integer in the field of degree n, which is a fixed integer x ~ 1 ~ 2, … , each of which can be any integer in the field, no matter what. In this way, by writing only a fixed integer aq1m aq2,... It is convenient to use symbols to express an ideal, that is, to use symbols to express an ideal, that is, to use symbols to express an ideal. As the symbol of the ideal. The order of the elements in the symbol is not important.

Now it is necessary to define the ideal "multiplication" of the ∶ two ideals (axi1, … , (bounded 1,... , Backn) is the product of the symbol (a_1b_1,... A_2b_2,... A_nb_n), in which there appears all possible products a_1b_1 obtained by multiplying an integer in the first symbol by an integer in the second symbol, and so on. For example,

(for an n-th field) it is always possible to reduce any such product symbol to a symbol containing up to n integers.

It can be seen that the work done by Dade King requires deep insight and a talented mind, and his abstract ability is far beyond the ordinary good mathematical mind. Dedkin always relied on his own mind, rather than on clever symbolic representations and skillful use of formulas to move forward. If there has ever been a person who put concepts into mathematics, Dedkin is one. He prefers creative ideas to boring symbols. This wisdom is now obvious. The longer mathematics exists, the more abstract it becomes-perhaps because of this, the more practical it becomes.

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

Welcome to subscribe "Shulou Technology Information " to get latest news, interesting things and hot topics in the IT industry, and controls the hottest and latest Internet news, technology news and IT industry trends.