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This article comes from Weixin Official Accounts: Fanpu (ID: fanpu2019), author: Zhang Hezhi

For a long time, people equated "number" with "real number." Real numbers rule the mathematical world like the sun. Renaissance algebraic mathematicians introduced complex numbers to solve equations. But even natural constructions like complex numbers took centuries to be accepted by the mathematical community. The status of real numbers seems unquestionable. By the end of the 19th century and the beginning of the 20th century, mathematicians were surprised to discover that the complete fields contained were not necessarily, but possibly, progressions. Like the stars, it is more like the moon: although the moon is the brightest in the night sky and often overshadows the stars, the existence of stars also reminds us that there is a farther space in this universe waiting to be explored.

God created the whole numbers, the rest is man's work.

- Leopold Kronecker

Motivation for the Introduction of Numbers

In fact, the number entered is not a symbol, but represents a prime number. The rational number field can be extended to the real number field, but this extension is not unique. The number of entries mentioned above means that for any prime number, it can be extended to the number field. Real numbers come from fractional expansions of rational numbers, while progressions come from progressive expansions of rational numbers. Although decimals are written in different bases, they are essentially different from progressions: decimal expansion defaults to progressively smaller, while progressions default to progressively smaller. We will explain this later. As shown below, real numbers and progressions have the same status.

Both real numbers and progressions contain rational numbers, and they are juxtaposed. Progressions were first introduced by the German mathematician Kurt Hensel, and before him Ernst Kummer had implicitly used this wonderful number. Like Kummer, Hensel's original work is hard to read. His article was published in 1897, when the concept of a field was only four years old: in 1893, Heinrich Martin Weber first defined a field as a set of operations with addition and multiplication, which can also be written as satisfying

Combination Law of Addition and Multiplication

commutative law of addition and multiplication

Addition and multiplication both have unit elements.

Each element has an additive inverse, which is

Every nonzero element has a multiplicative inverse, i.e.

Multiplication satisfies the distribution law for addition

The rational numbers and real numbers we are familiar with are fields. Weber defines it this way because he wants to include modular residues, such as arithmetic for seven days a week. If the condition of multiplicative inverse is removed, the above definition becomes a so-called commutative ring, and the most typical example is the integer ring.

The problem of number theory is usually related to, and if nonzero elements are allowed to have multiplicative inverses in, we get, this construction is called fractional field of. Since many of the results obtained in can be directly applied (e.g., a polynomial with leading coefficients has rational roots if and only if it has integer roots), we usually consider them together. But the properties of both objects are "bad." For example, we want to determine whether there is a rational number solution for some pair of non-zeros. It seems impossible to start. But if you want to determine whether there is a real root, it is simple: if there is one, there is a real solution, otherwise there is no real solution. If so, then it is a real solution. But if, then for any real number, is constant, so there is no real solution. Obviously, if there is a rational number solution, then there must be a real number solution, after all, but the reverse is not necessarily true. Does the existence of real solutions help rational solutions? The answer is yes, for which we need to define Hilbert symbols (yes "or", yes "and"):

To solve the problem of intelligible judgment, one needs to define Hilbert symbols for each prime number. This definition is also elementary, but slightly more cumbersome, and interested readers can consult Reference [1] for themselves, and we will not deal with this definition itself later. The point is that this definition can be calculated directly, so it is easy to judge. Mathematicians proved an astonishing theorem: there is a rational number solution if and only if it holds for all.

This theorem is indeed very convenient, but it raises a deeper question: since it can be interpreted as judging whether there is a real solution, does it also correspond to an extension field of if and only if the equation has a solution in this field? If this is true, then it seems that we can think of rational solutions as the "intersection" of solutions in all these fields.

Of course, intersection is not accurate. In conclusion, what we are looking for is exactly the field of progression, and all these sums together can be called corresponding "local fields." It is the "global domain."

The above theorem is actually about the correspondence between local and global. This sounds incredible. The domain has obviously become larger, but it has changed from a whole to a part. To explain this, we need to understand some geometry.

Analogy between integer rings and polynomial rings Mathematicians noticed similarities between number theory and geometry long before abstract ring theory was born. In particular, they are very similar to rings, for example, both rings can be divided by remainder, so they are Euclidean integral rings. Here is the polynomial ring of coefficients. Even if this coefficient field is replaced by other fields, there will be many similarities, but we need to use some analytical methods here, so complex numbers are the most convenient. Incidentally, their fractional fields and sum are similar. This means that nonzero polynomials are allowed to divide. The elements of can be thought of as meromorphic functions on: their denominators are not necessarily nonzero at individual points, so these functions have poles that tend to infinity, but these points are discrete and easy to handle. For me, part obviously means any one of these points. These meromorphic functions can be expanded into Laurent series at any point, just as holomorphic functions (analytic everywhere) can be expanded into Taylor series at any point, except that Laurent series allow such terms to exist. For example, near a point, you can expand

form. At any point we can define the order of a meromorphic function as the degree of the leftmost term of its Laurent expansion. For example, the order of this function at this point is. Similar expansions can also be performed in. Generally speaking, for some rational number, we can write it in the form of, where is different prime numbers, is an integer, positive or negative. Definition. Is there any way we can expand it into something similar

What about the form? The answer is yes, and you can formalize progress.

Why can I write like this? For ordinary real division, the quotient will be longer and longer after the decimal point, because the later we default the number, the smaller its "size", so we can write such an infinite decimal. But to make the above expansion, in fact, the default sequence will become smaller and smaller. We write it first, so we only need to calculate and finally move the whole place by one bit. calculated as follows

The careful reader will find that the reason why such divisions yield a quotient of one digit at each step depends on the fact that it is a field, so that numbers that are not prime numbers, not fields, cannot be expanded in this way.

That'll be considered out.

Now, relying entirely on analogy, we have this expansion. For any prime number, we call such an expansion an advance expansion. This expansion is very similar to the decimal representation, which explains its name. But this is purely formal. We also need to explain three questions:

The Laurent expansion of rational functions at a certain point is obviously related to "local," but why is the development of rational numbers at prime numbers also called local?

Why is it partial?

How exactly do we define progress? In other words, how to define it?

Why local? We need to associate the dot in with so we know what the dot means for us. For this we need ideal concepts. For a commutative ring, an ideal is a proper subset satisfying the following properties:

Closed for addition and subtraction;

In other words, after multiplying the element by any element in, the result is still in.

This definition was originally proposed by Ernst Eduard Kummer and Julius Wilhelm Richard Dedekind to solve the problem of the failure of element decomposition in algebraic number fields (which is why it is called an ideal: a subset of very "ideals"), but algebraic geometers have found its geometric meaning. We use it to represent the smallest ideal contained in (that is, the ideal generated by). This is a maximal ideal, that is, it is not a proper subset of any ideal. In fact, for any point in, is a maximal ideal. And conversely, all maximal ideals in are of the form. Therefore, the points correspond to the maximal ideals of. Thus we can consider as its point the maximal ideal of, which is exactly the ideal of all forms.

Such simple analogies cannot really be called "geometry." It was not until Alexander Grothendieck creatively proposed the theory of schemes that algebraic geometry and number theory could be truly unified. In this theory, the prime ideal of a ring (which is not needed here) is called a point, while the maximal ideal is a closed point. This theory requires more sophisticated background knowledge, so I won't introduce it in this article. In short, the Laurent expansions and progression openings we used above are closed points corresponding to two rings. If you accept this setting, you will find that there is nothing wrong with the term "local."

So what is the expansion in, which is decimal expansion? It is actually the Laurent expansion of the corresponding rational function at infinity. as shown

img Any point on the complex plane can correspond to a point on the sphere, just connect the top of the sphere to a point on the complex plane, and the line segment must intersect a point on the sphere. This establishes a one-to-one correspondence between the complex plane and the sphere (except at the apex). And if you approach infinity in any direction on the complex plane, you must approach the vertex if you convert to the sphere. Thus we can regard this sphere as an extension of is and call it a Riemann sphere, denoted.

Now to do Laurent expansion of rational functions at infinity is to treat rational functions in as functions of yes and then do Laurent expansion at infinity. that is

Because of this similarity, the discriminant we defined above is written.

To define, we first have to know what it is. Logically, the first definition should be natural number, then natural number but how does each step come about? It is defined by the Peano axiom, that is, from the beginning, it stipulates that every number has a successor number, so mathematical induction can be used. And then we're going to get, what do we do? Intuitively, defining integers allows for the existence of negative numbers. But what exactly are negative numbers? For example, it actually is, or it could be. So if we want to define it, an integer is actually an equivalence class in, that is, at the time, we specified equivalence relations. This can be defined as the set of all equivalence classes. Of course it is a subset of, because the natural numbers are equivalent to this equivalence class. A similar approach can be constructed: since fractions are allowed to exist, and if, then, so we define where then. Integers can also be equated with equivalence classes, so they are also subsets of. The above two extensions allow some new operation and then construct it by taking the equivalence class.

What operations are allowed? The answer is limit. In hindsight, the sequence is as follows

The limit of 0 is 0, but now we only have 0, so we can only say that this sequence is not convergent in 0. If all sequences like this converge to a single number, that must be it. But not all sequences converge, for example

So we need to restrict the sequence and then take some equivalence class. The restricted sequence is called a Cauchy sequence and is defined as follows: For rational sequences, there exists one for every such sequence such that there exists one if. Intuitively, it requires that the tail of the sequence swing toward. It is not difficult to prove that all sequences converging to rational numbers are Cauchy sequences, so this can be said to be a natural generalization of moderately convergent sequences. Of course, it is possible for two Cauchy sequences to converge to the same number, so we also need equivalence relations if and only if. Thus all equivalence classes in the set of all Cauchy columns are defined as. All rational numbers are equivalent to the Cauchy series of constants, and therefore are subsets of. This may also explain a problem that is difficult for laymen to solve. It's actually Cauchy, but it's Cauchy. And their difference is a sequence that tends to be so two Cauchy columns are equivalent.

However, we should note that Cauchy's definition depends on. Of course, the definition here is absolute value in the ordinary sense. Absolute values represent the distance between two numbers. In the middle, it gets smaller and smaller. But we see that in the progress above, the smaller and smaller is, which suggests that we should change the definition of this distance, and we call this new distance for the time being, the progress quantity. The bigger we need, the smaller we need, so a natural definition is. In fact, the base number does not have to be, any number greater than can be taken (they decide that Cauchy is exactly the same), the reason is only for convenience. Of course, the distance is not arbitrary, and the function needs to satisfy three properties to be called a metric function (which actually defines the norm on the field):

If and only if;

;

The triangle rule is that the sum of the two sides is not less than the third side.

Thus, as long as there is a distance function, we can define Cauchy columns, and we can define new fields. This process is called completion because we call any domain in which Cauchy sequences converge complete. So to summarize, that's absolute metric completion of 0 and progress metric completion of 0 defined as 0 is the progression field that we want. We can even define a similar distance, and the completions obtained are the formal Laurent series fields and. A formal Laurent series is an expression in the form of a Laurent series without dealing with convergence problems. Then by Laurent expansion, embedded in these forms Laurent series field as subsets.

However, we do not call it a local field, which is for other reasons, unrelated to this paper. We can see that these embedding relations are very similar to the progression.

Since Cauchy can be defined for any given measure, is there any other way to define distance besides absolute values and progress quantities? The answer is no. In, any measure satisfying the above three properties is equivalent to an absolute value or a certain progress quantity. That is to say, all the completion schemes mentioned above are complete. We don't always consider Cauchy columns when calculating real numbers, but decimal expansions are more common; similarly, when actually calculating progressions, progression expansions are more common.

Using the above construction, we can prove if and only if the equation has a solution in. So the theorem we mentioned at the beginning can be stated as: There is a solution in if and only if there is a solution in all and.

It is natural to ask whether, given an arbitrary polynomial equation, the conditions for its existence and comprehensibility are equivalent to the existence of real solutions and all kinds of exponential solutions. The answer is no, there are many polynomials do not hold this conclusion. This piqued mathematicians 'curiosity: Which polynomials have similar properties? We call this direction the part-whole principle, and the new knowledge it has generated continues to nourish the whole study of number theory to this day.

What does it have to do with reality? Indeed, number theory was a discipline very far away from the real world. In recent years, a number theory has been applied to cryptography. There is not much work done to apply it directly to physics to describe the real world and be accepted by most physicists.

This is actually very strange logically. But why is it that all of our physical theories today are described in terms of their algebraic closures? There is no logical distinction between progressive numbers and real numbers. They can both be derived, integrated, and most analytical tools you can think of can be applied equally to them. So why do we live in the real world instead of the progressive world?

Someone really thought of this possibility. In string theory, the world surface swept by a string is described by a one-dimensional complex manifold (i.e., Riemann surface), but if Riemann surface is replaced by the corresponding concept in geometry, a string theory can also be created, called string theory. At present, the research results in this area are still in the toy stage. But that doesn't stop us from being curious. After all, we look up at the night sky only because the stars are beautiful.

references

[1]Kazuya Kato, Nobushige Kurokawa, Takeshi Saito. Number theory I-Fermat's dream and class field theory.

[2] Neal Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions.

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