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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), by Kasper M ü ller, translated by Xu Zhaoqing

Galois, a mathematical genius, died in a duel at the age of 20, ending his short life, and the quintessence of his thoughts will always flow in the long river of history.

In the early morning of May 30, 1832, only 20-year-old Evariste variste Galois was injured and fell on the dewy grass with a gunshot. One of the most fascinating and mysterious figures in history is about to come to an end.

Galois, Photo Source: Wikimedia Commons introduction this is a story about love and math, about a very smart young man. His scribbled manuscripts opened one of the most beautiful and interesting areas of mathematics and triggered a revolution in how we think about equations. Not only did he solve a 350-year-old problem, but his theory also provided answers to several unanswered questions in 2000. We'll talk about that later.

More specifically, Galois considered the problem of finding the roots of polynomials. (translator's note: the root of a polynomial is also called the solution of a polynomial, even if the value of x of the polynomial p (x) function is zero.)

At that time, mathematicians already knew that there was no general formula for finding roots for polynomials of degree 5 or more. (for the formula here, we mean taking the n-th root and applying four operations. This concept is also known as root solvability, which is referred to as solvable in this paper. However, Galois wants to understand why some higher order polynomials are root solvable while others are unsolvable. (translator's note: here the reader can use the root formula of quadratic polynomial as an example to understand the concept of root solvability. )

For example, the equation x5-1x0 is solvable, and we call these solutions quintic unit roots. These solutions are beautifully distributed evenly on the unit circle of the complex plane, and they are also the vertices of a regular Pentagon, that is, five quintic unit roots.

So some polynomial equations of order d (where d ≥ 5) are actually solvable! The problem solved by Galois theory is precisely why this is the case and which equations are root solvable, rather than just knowing that some equations are unsolvable.

The fact that some polynomial equations are unsolvable is proved by another genius, the young Norwegian mathematician Nils Henrik Niels Henrik Abel. In fact, several great mathematicians, such as Ruffini and Augustin-Louis Cauchy, also contributed to this, but no one came up with a theory close to Galois, and no one could explain for sure why.

In this article, we will first understand the general situation of history and the life of Galois, and then briefly introduce his mysterious death at the age of 20. After that, we will see the whole picture of its beautiful mathematical theory and discuss why it is so elegant.

Although an article cannot cover all of Galois's theory, I hope to show you part of its elegance and beauty. I hope it inspires you to learn and explore.

Galois was born on October 25, 1811. He was interested in mathematics very early, and at the age of 14, he found the book É l é ments de G é m é trie by Adrien-Marie Legendre. It is said that he read the book "like a novel" and mastered it the first time.

At the age of 15, he may have been greatly inspired by the fact that he began to read Lagrange's papers.

Although Galois studies hard in his own time, he has little motivation in class.

In 1828 and 1829, he was twice rejected by the Paris Comprehensive Institute of Technology, which was the most prestigious mathematics college in France at that time. The first time was because he was partial to the subject, and the second time because he failed the oral exam. It is said that he screwed up the oral exam. (translator's note: the Paris Comprehensive Institute of Technology is considered to be the top engineer university in France and is regarded as the pinnacle of the French elite education model. )

From this moment on, time flies, and in 1829 Galois published a paper on continued fractions, and at about the same time, he contributed some papers on polynomial equations. The reviewer was one of the greatest mathematicians of his time: Augustine Louis Cauchy.

But although Cauchy suggested that Galois submit the article to the French Academy of Sciences to participate in the Academy Award (Grand Prix), he did not publish Galois's paper.

To this day, no one knows why Cauchy didn't publish it. It was said that he recognized the importance of Galois's ideas, but suggested that Galois make some edits before publication. Some people say that political factors have played a role. (obviously, the political views of Cauchy and Galois clashed, which was a big deal at the time.)

On July 28, 1829, Galois's father died. Galois had a very close relationship with his father, so it was a heavy blow in his life.

In 1830, at Cauchy's suggestion, Galois submitted a paper on equation theory to another mathematical guru, Joseph Joseph Fourier. Unfortunately, Fourier died soon after, and Galois's paper was lost.

It was a setback for Galois, of course, but he didn't give up easily. Later that year, he published three papers. One of them summarized the content of what became known as Galois theory, and the other first studied the mathematical concept we now call finite field (Finite field), which was later very important in the field of number theory.

In order to understand Galois's situation and life, we need to understand what happened in France. At that time, in the middle of the French July Revolution, also known as the second French Revolution, Galois not only participated in the revolution, but also participated in the battle and debate. He joined the street riots and spent all his time on math and politics.

The mystery of Galois's death in the years after his father's death, Galois became more and more violent and he was arrested many times. In January 1831, Galois tried again to publish his theory, but the great mathematician Simon é on Denis Poisson called his work "inexplicable".

Galois was in prison and was angry at Poisson's rejection. But for some reason, this time he took the criticism very seriously and began to sort out his work and write his statement more carefully.

Galois was released on April 29, 1832. Soon after, he took part in a duel.

There is a lot of speculation about the famous duel. A letter written by Galois five days before the duel indicates that he is in love, and the duel is for his lover.

The night before the duel, Galois, convinced that he was going to die, stayed up all night and wrote his greatest contribution to mathematics: the famous letter expressing his views to Auguste Chevalier, and three attached manuscripts.

The last page of Galois's manuscript. Photo Source: Wikimedia Commons mathematician Hermann Weyl said of the manuscript.

"judging from the novelty and profundity of the ideas contained in this letter, it is perhaps the richest article in the whole human literature."

This is the famous saying of a great man.

In the early morning of May 30, 1832, Galois was shot in the abdomen and was later abandoned by his opponent.

The next morning, the 20-year-old Galois died.

Later in the story in 1843, Joseph Liouville (Joseph Liouville) reviewed Galois's manuscript and declared it correct. The paper was finally published in 1846, 14 years after Galois's death.

However, it took longer for this theory to become popular among mathematicians, and people really understood its secret.

In fact, Liouville completely missed the theoretical core of Galois method-Group. It was not until the turn of the century that Galois theory was fully understood and established as the core part of abstract algebra (Abstract algebra). It took nearly a hundred years for this theory to become the standard content of algebra courses.

The most famous part of Galois's manuscripts is to prove that the root formula of quintic polynomials does not exist-that is, quintic and higher order polynomial equations usually cannot be solved by roots.

As mentioned above, Abel proved that the "quintic formula" of root solution is impossible in 1824, but Galois carried out more in-depth theoretical research and put forward the current Galois theory.

This theory can be used to determine whether an arbitrary polynomial equation has a root solution.

Galois was the first to invent the word "group", using (almost) the same definition that we use today in different universities and colleges. He proposed the concepts of normal subgroup (Normal subgroup) and finite field, which we will discuss later.

In essence, Galois is one of the pioneers in the field of modern group theory and abstract algebra.

Group theory is the study of symmetrical mathematics, which is widely used in many disciplines of mathematics and physics. Abstract algebra is also called "the language of modern mathematics".

I remember clearly that before I studied Galois theory, I had taken many courses in abstract algebra, such as group theory (Group Theory), ring theory and ideal (Ring and Ideal Theory), domain theory (Field Theory) and module theory (Module Theory), all of which were very abstract.

Then I learned Galois theory, and a lot of what I learned before, especially group theory and domain theory, have been applied. Finally, I can use all these abstract mathematical objects to prove why some particular polynomial equations do not have root solutions, and these are not all Galois theories.

That's why I think Galois's theory is wonderful.

Galois theory Galois theory connects two subfields of abstract algebra-group theory and field theory.

As mentioned earlier, Galois theory was born out of the following question:

For a polynomial equation of degree 5 or higher, is there a formula that can express all the roots of the equation by using the coefficients of the polynomial, the commonly used algebraic operations (addition, subtraction, multiplication, division) and roots (square root, cubic root, etc.)?

Although the Abel-Ruffini Theorem (The Abel-Ruffini theorem) provides a counterexample to prove that there is a polynomial equation so that such an expression does not exist, Galois's theory can explain why some equations, including all quartic and lower equations, are possible to find root solutions, and why many quintic and higher equations do not have root solutions. Thus it provides a more complete and clearer answer to the previous question.

Modern Galois theory uses the language of groups and domains, so I will try to explain Galois theory while avoiding too much other knowledge, but for completeness, we will briefly introduce these mathematical concepts.

Group theory is the study of symmetry.

Imagine a square: the square has a certain degree of symmetry-if you rotate 90 degrees, it looks the same, 180 degrees and 270 degrees; of course, if you rotate 360 degrees, it will return to its original state.

In order to record it, we can imagine that all four corners of the square are marked so that we know how to transform.

There is also a reflective symmetry, such as selecting an axis, or a line, passing through the middle of a square and dividing it into two rectangles of equal size. You can flip the square along this line, it still looks the same, but this transformation is different from rotation.

The last one is trivial symmetry (nothing changes).

Every symmetry has an antisymmetry: for example, if you rotate 90 degrees clockwise and then 90 degrees counterclockwise, the two transformations cancel each other out and are finally equivalent to trivial symmetry.

This concept can be generalized by algebraic methods.

A group G consists of a set and an operation that satisfy the following conditions:

1. For the elements g, h in the two groups, after the operation, we will get the element g in the group.

two。 There is a unit e so that any element g does not change after its operation, g*e=e*g=g

3. For any element g, there is an inverse a that makes g*a=a*g=e.

In the above example, the element in the group is the transformation itself. For example, the rotation of 90 degrees and the reflection transformation mentioned above are both elements in the group. We record the rotation of 90 degrees as σ and the reflection transformation as τ.

The operation of this group is the composition of transformations. So we can get σ * τ, that is, first flip along the axis of symmetry, and then rotate 90 degrees. But we can notice that σ * τ ≠ τ * σ, so the order of element operations in the group is very important. (translator's note: we might as well assume that the rotation rotates clockwise and the four corners of the square are labeled, so that the reader can verify by drawing that the result of flip and rotation is different from that of rotation and flip. )

So the concept of group is a way to abstract symmetry. In fact, there are so many groups of abstract transformations that we don't even know how to visualize some of them.

But one of the simplest groups is familiar: sets containing all integers and addition operations form a group.

When we add two integers, we get the third integer (this set is stable for addition). The unit is 0, because for any integer k, 0+k=k+0=k, and the inverse is-k, k + (- k) = 0.

So, it's a group. But what symmetry does the set of integers and the group of addition operations reflect? The answer is translational symmetry. Plus an integer k can be regarded as the translation distance k along the number axis, and the positive and negative represent the direction.

The subgroup H of group G is generally recorded as H < G, which means that it is a subset of G and also forms a group. For example, a set of even numbers is a subgroup of an integer addition group.

Domain theory in mathematics, domain is a special ring. You can think of a field as a set of two operations, which are usually recorded as addition and multiplication, that is, + and *. The addition and multiplication here may not be commonly used operations, they depend on the definition of the field, but you will see why this token makes sense. There is a zero element so that for any element a, an is zero.

Moreover, the set is a group for the defined addition +, and the set\ {0} is also a group for the defined multiplication *. Not only that, the two operations satisfy the distribution law, a * (Backc) = a*b+a*c, where the multiplication and addition operations are operations defined in the field.

Other well-known properties are the existence of unit 1 in the field and the commutative law of operations, a+b=b+a, a*b=b*a.

These two properties may seem familiar. This is true, because the familiar real numbers and complex numbers are fields and satisfy these properties.

If you understand modular operations, you will know that integers are a field for any prime p, (often recorded as), and a finite field! This is one of Galois's discoveries.

So, a field is a collection of "numbers", and we can perform four operations in the field with the usual rules, and they all have inverse. (except for the multiplicative inverse of zero, because it is still impossible to divide by zero in the field. )

Galois theory is concerned with the extension of the field of rational numbers (representing rational numbers, that is, fractions in which the numerator denominator is all integers) and the subfields of complex fields, which contain only a finite number of irrational numbers.

We must add at least one irrational number to the rational number field to get such a field in the middle. So what are these domains?

We know that it is not a rational number because it cannot be written as a fraction in which the numerator denominator is an integer. However, we can add it to rational numbers. Of course, in order to get a field, we need to add a lot of other elements, such as -, that is, its additive inverse. In fact, we need all numbers in the form axib, where an and b are rational numbers.

We call this set the generated extension field added in, or a single extension field, recorded as. It can be verified that every non-zero element in the extension field has addition inverse and multiplication inverse.

More generally, we can think of (α) as the smallest field containing all rational numbers and α. If α is a rational number, then we get a trivial extension.

Before we can discuss the beauty of Galois theory, we also need to know what the Splitting feild is. But it's very simple.

Consider a polynomial f of degree n whose coefficients are all rational. We know from the basic theorem of algebra that the polynomial f of degree n happens to have n complex roots (the multiplicity of roots is included).

So we can consider the domain extension based on which all roots of the polynomial f are included. The minimum domain that satisfies the condition is called the splitting field of the polynomial f, because we can decompose the polynomial f factor in this field.

The last concept is the automorphism (Automorphism) of field K. This is an ingenious word used to denote a replacement that maintains a structure in a domain. If σ is an automorphism of K, then

σ (x) = σ (x) + σ (y), σ (x) = σ (x) * σ (y)

And σ is a bijection, that is, the mapping is both a unijective and a surjective.

Suppose that the field K is an extended field of the field F, that is to say, F is a subdomain of K. We can consider the automorphism σ on the K of the fixed field F, and for any element x of the field F, σ (x) = x.

The basic theorem of Galois theory for a given polynomial, different algebraic equations can connect different roots. In this paper, the algebraic equation refers to the polynomial equation with rational coefficients. )

The main idea of Galois theory is to consider the replacement of roots, so that the algebraic equation that was originally satisfied is still valid after the replacement.

The group formed by these permutations is called the Galois group of the polynomial.

For example, let's consider f (x) = x2-2x-1. The two roots of this polynomial are marked as α = 1 +, β = 1 -.

The algebraic equation satisfied by two roots is

α + β = 2

α * β =-1

It is not difficult to see that it is still true after exchanging α and β in the two equations. In fact, all algebraic equations for α and β are true after transformation.

A popular way of understanding is that, in a certain sense, rational numbers cannot tell the difference between 1 + and 1 -.

"and-are the same anomalies for rational numbers."

So, the Galois group of f has two elements, a trivial permutation and a commutative permutation of two roots, that is, changing 1 + into 1 -, and vice versa, and fixing other rational numbers. This is exactly the cyclic group of order 2, isomorphic to. (in higher mathematics terms, this means "two groups are the same". )

In modern language, we can consider the splitting field K of f, assume that there are different roots, and define the Galois group of f as the automorphism group of all K with fixed rational numbers.

We generally call this automorphism group Gal (K /), where K / F, in this example, means that the field extension K is based on the field F, and the automorphism can fix the field F.

Or we can put it another way, this automorphism group contains all permutations that satisfy the following conditions: after the permutation acts on the root of the polynomial, the algebraic equation satisfied by the root of the original polynomial is still valid.

For the previous example, we have this isomorphic relation, Gal (() /).

More generally, we define the Galois group of the field extension K based on the field F as the automorphism group of K that can fix the field F.

Under this naming rule, the Galois group of the polynomial f refers to the Galois group whose domain is split. (as mentioned earlier, a split domain refers to a field extension based on all roots of a polynomial f. )

For any field K and can fix the automorphism σ of field F (usually denoted as σ ∈ Aut (K / F)), if a polynomial with any coefficient has a root α, then there is also a root σ (α). Therefore, such an automorphism will indeed be based on the field F, replacing the root of the minimum polynomial of α.

In addition, with a similar idea, we can prove that if a complex a+bi is a root of a real coefficient polynomial f, then its complex conjugate a-bi is also the root of a polynomial f.

This is because there is an automorphism that can replace I and-I. Therefore, σ (a+bi) = σ (a) + σ (bi) = aquib σ (I) = a-bi.

The basic theorem of Galois theory is that the sum of subgroups of Galois group Gal (K /) is one-to-one corresponding to the intermediate field of K.

In fact, this theorem is not only that, given an intermediate field, ⊂ L ⊂ K, the corresponding subgroup H < Gal (K /) exactly contains those fixed L automorphisms.

The solvable group Galois himself understood and studied in that famous manuscript at that time, considering a polynomial f, if the Galois group of f is a solvable group (Solvable group), then the polynomial is root solvable, and vice versa.

Of course, I also need to tell you what solvability means to a group.

Consider a group G and its subgroups H, H < G. If the following condition holds: for the element h in H, and the element g and its inverse an in group G, and the element g*h*a ∈ H, we call H a normal subgroup of G.

This means that H is invariant under the action of group G, or under the conjugation of group G elements.

More generally, through the elements of the normal subgroup H and the group G, we can construct an equivalent relation. This requires the use of Cosets theory, but we do not assume that the reader is familiar with this, which is outside the scope of our article. So we say here that this equivalence relation can construct a new group.

When we give an integer modulo n, we can construct a cyclic group by equating the integer multiples of all n to 0; this is what happened above. Where is a normal subgroup because it is an abelian group (a+b=b+a), and any subgroup of an Abelian group is a normal subgroup.

You can also understand it in a more abstract way, considering any normal subgroup H < G, and the corresponding group of modular operations is denoted as G / H, which is called G module H.

Furthermore, if group G contains a nested normal subgroup chain, {e} = H0 < H1 < H2 < … < Hk=G, such that for any index I ∈ {0,1,2, … If Hi+1/Hi is abelian, then we say the group G is solvable.

This sums up how Galois theory is related to the solvability of polynomials.

We can find an example of a rational coefficient polynomial and prove that it is not root solvable by studying its corresponding Galois group.

For example, the polynomial f (x) = X5-6x+3, we can use the mean theorem and some techniques to prove that its corresponding Galois group is a five-letter permutation group S5. This is not a solvable group, so f is not radically solvable.

The beauty of Galois's theory is that we can associate each polynomial with a group of algebraic information that maintains its roots. By studying this group, we can transform the algebraic information into the world of polynomials.

I mentioned earlier that we can use this theory to prove some very old problems.

As a by-product of Galois theory, the problems of "cubic multiplication" (Doubling the cube) and "turning a circle into a square" (Squaring the circle) finally proved impossible. They are all related to the extension of the rational number field mentioned earlier.

For example, the problem of turning a circle into a square is equivalent to showing that π is the root of a rational coefficient polynomial, but this is impossible. Because π is a transcendental number, it is not in the extension of any finite algebraic field.

The same is true for cubic multiplication, but we need to consider the number of domain expansions that join the cubic root of 2. If you are interested in this problem, you can try it yourself.

Evariste Galois is without a doubt a first-rate genius. The times and environment have brought him a lot of difficulties, and his arbitrariness is considered unconventional in the field of mathematics, and to some extent, it is not accepted now, because mathematics needs to be very accurate and careful to avoid ambiguity. Mathematicians often use the term "rigorousness" to describe this requirement.

But that doesn't mean his theory is incorrect. Galois theory is correct and beautiful! Now, it is used in many different fields of mathematics, including Andrew Andrew Wiles's proof of Fermat's great theorem and algebraic number theory.

The idea of using groups to represent another structure is wonderful. This idea is now applied in many fields, such as in algebraic topology (Algebraic topology), we can study a group to get the information of topological space; in algebraic geometry (Algebraic geometry), we can study the solution set of polynomials by using ring theory and ideal theory; points on elliptic curves form a group, and so on.

Dear readers, if you read here, I hope you enjoy this journey about Galois.

Thank you for reading.

This article is translated from Kasper M ü ller, For the Love of Mathematics, original address: https://www.cantorsparadise.com/for-the-love-of-mathematics-84bf86a8ae09

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