(5,000 words chapter, there is a lot of popular science content in the history of mathematics in the second half, I originally wanted to skip it directly, after all, the content of nearly 2,000 words may affect everyone's reading experience, but think about it carefully, let's keep it, everyone thinks it's interesting, then take a closer look, if you think it's boring, you can skip it directly)

Pang Xuelin certainly understands why Liu Tingbo is so excited, in the domestic mathematics community, Peking University dominates, Fudan Tsinghua University is slightly inferior, and the rest of the mathematics departments of University of Science and Technology, Jiaotong University, and Nanjing University are almost all ranked above Jiang University.

In recent years, the development of the Department of Mathematics of Jiangxi University in China has been tepid, and there are almost no particularly influential mathematicians, and Pang Xuelin is already one of the best.

If Pang Xuelin proves the BSD conjecture this time, his academic achievements will be enough to stand alone in the domestic mathematical community.

At that time, as long as Pang Xuelin is still in Jiang University, it will be a towering tree that can compete with the School of Mathematics of Peking University.

Although there is still a certain gap between the Department of Mathematics of Mizuki and Peking University, it is no longer under Fudan, an old mathematics school in China.

Pang Xuelin's existence will play a similar role to Yau Chengtong.

And this BSD conjecture report will be the first shot fired by the School of Mathematics of Jiangxi University in the international mathematical community!

In the next few days, the dark tide of the mathematical community surged, and Pang Xuelin's paper hanging on arXiv entered the field of vision of most mathematicians who paid attention to this field.

Even in some university forums in China, there has been a gradual discussion about Pang Xuelin's proof of the BSD conjecture.

"Did you hear that? The BSD conjecture was proved by a professor named Pang Xuelin at Jiang University! ”

"Who is Pang Xuelin, I haven't heard of it!"

"Didn't you watch the news? A new genius mathematician in China in recent years, only 22 years old after graduating with a doctorate and publishing a top journal paper, under the tutelage of Professor Tao Zhexuan, returned to China this year and directly entered Jiang University as a professor and doctoral supervisor. ”

"It's pretty awesome to look at the resume, but BSD guesses that it's one of the seven major problems of the millennium, so it doesn't feel very reliable!"

"I heard from my professor that the paper is still in the peer review stage, and it is said that it has been recognized by many big bulls."

"Who are you teaching?"

"Akshai Venkatsh! He is a 2018 Fields Medal winner whose main research areas are analytic number theory, topology, and representation theory. ”

", it turned out to be Stanford's senior brother, senior brother is above, please accept the younger brother!"

……

There are endless discussions like this, and even on the campus of Jiang University, there are similar rumors.

On the morning of September 11, 2019, today is the first class Pang Xuelin has taken since joining Jiang University.

At seven o'clock in the morning, he arrived at the lecture hall early.

The class time was eight o'clock, and when Pang Xuelin arrived, the classroom was empty, Pang Xuelin found a place to sit down in the first row, then turned on the computer, and continued to write the PPT of the report meeting.

After about twenty minutes, Pang Xuelin suddenly felt a fragrant breeze from his nose, raised his head, and saw a beautiful girl with a tall figure, a high nose and deep eyes, a delicate face, and a little like a mixed-race child sitting down in the first row of the classroom.

The classroom they were in was a large lecture hall, and the two sat in the first row across the aisle, Pang Xuelin sat on the left side of the aisle, and the girl sat on the other side across the aisle.

The two looked at each other, and the girl smiled lightly at Pang Xuelin and said early.

Pang Xuelin nodded at her and did not speak.

After a while, the girl saw Pang Xuelin crackling on the keyboard, so she got up and walked to Pang Xuelin's side, and said with some curiosity: "Classmate, what are you doing here?" “

"Make a PowerPoint!"

"What are you going to do with this report? By the way, why haven't I seen you before, and you're here to skip class? Today is Pang Shen's first class, and many students from other majors have come to worship Xueshen. “

Pang Xuelin was slightly stunned, raised his head and said, "What about you?" ”

The girl said: "I am from the Department of Mathematics, and "Abstract Algebra" is a basic course, of course, there is no need to skip the class." “

Pang Xuelin couldn't help but be stunned, the ratio of male and female students in the Department of Mathematics has always been very different, such a beautiful girl actually chose to study mathematics, and the boys in their class are blessed.

At this time, the girl said, "But I can see idols today, it's exciting to think about." Hee-hee, have you heard a piece of news, and now the forums are saying that Pang Shen proved the BSD conjecture, which is one of the seven problems of the millennium, and it is said that the Clay Institute has also offered a million dollar in prize money. “

"I don't think Professor Pang is doing research for that money!"

The girl gave him a blank look and said, "Of course not, but don't you think Pang Shen is very powerful?" He is only 23 years old, and he is the same age as us, and he is already a full professor in our school. ”

Pang Xuelin nodded and said, "That's why you should work harder!" ”

The girl raised her little hand and said, "Yes, you must work harder, my goal is to be admitted to Pang Shen's graduate school in two years, hehe!" ”

Pang Xuelin glanced at this girl a little speechlessly, this girl is quite eye-catching, but why doesn't she look smart?

In the future, if she really applies for her own graduate school, she may have to think about it, such a twittering apprentice, it is estimated that she will be annoyed to death by her.

But fortunately, the girl saw Pang Xuelin turn his attention back to the computer, and she was embarrassed to continue to disturb Pang Xuelin, so she sat aside and read the book intently.

After a while, people began to come in one after another, and the girl seemed to be quite well-known, perhaps because of her big personality, and many people came to greet her.

Especially after the arrival of her best friend, the girl returned to the essence of chattering, chattering non-stop.

Perhaps it was really for the sake of coming to worship the God of Learning, but today's lecture hall of more than 300 people was packed, and even the aisles were full of people.

Time passed minute by minute, and before I knew it, it was eight o'clock.

"Hey, it's time, why hasn't Pang Shen come yet."

"Wouldn't it, Pang Shen was late for the first day of class?"

"It is said that abstract algebra is particularly difficult, and there are many concepts, I hope Pang Shen will not follow the book in class, otherwise the final exam will be troublesome."

……

The students talked a lot, Pang Xuelin's face did not change, closed the notebook, walked to the podium, put on the microphone used for sound amplification, and said lightly: "Hello everyone, I am the Pang Shen Pang Xuelin in everyone's mouth, and I am responsible for the teaching of the "Abstract Algebra" course this semester. In order to ensure the quality of teaching, please try to keep quiet, I will reserve 20 minutes of question time before the end of class, and you can ask any questions then! “

There was an uproar, and the classroom suddenly became noisy, and no one expected that Pang Xuelin was sitting in the first row of the classroom.

The students sitting next to him were even more dumbfounded, they didn't expect that this boy who looked a little handsome was the legendary Professor Pang?

Especially the beautiful girl who had talked to Pang Xuelin, she looked sluggish, and then cried to her best friend next to her: "It's over, Xiaoyue, I was talking to Professor Pang just now, and I said that I was going to apply for his graduate school, do you think he will leave a bad impression on me?" “

Xiao Yue smiled: "Ai Ai, you are so beautiful, how could Professor Pang refuse you to be his student?" ”

"Okay, you dare to say that I have good looks and no IQ!"

"Shhh......h Professor Pang is on the podium, don't you really want him to make a bad impression of you? In other words, Professor Pang is really handsome, especially on the podium, he is very charming! “

"You...... Hum! ”

Ai Ai glared at her best friend, and when she saw Pang Xuelin's gaze sweeping over, she quickly shrank her head, and the corners of her mouth turned up slightly: "Pang Shen is really a little handsome!" ”

Pang Xuelin didn't care about the noise in the audience, and said: "Okay, let's get to the point, the abstract algebra course that everyone will learn this semester is also called modern algebra, which was born in the nineteenth century. It contains many branches such as group theory, ring theory, Galois theory, lattice theory, linear algebra, etc., and combines with other branches of mathematics to produce new mathematical disciplines such as algebraic geometry, algebraic number theory, algebraic topology, and topological groups, and abstract algebra is also one of the foundations of modern computer theory. ”

"To some extent, abstract algebra has become the lingua franca of much of contemporary mathematics. Therefore, although abstract algebra is a basic course, it plays a very important role in the future in both academic and application fields. “

"However, abstract algebra is a complex concept and very difficult to understand, and if you start directly with the concept according to the textbook, I think everyone will feel very confused. However, since abstract algebra has the word algebra, it must be related to solving equations, and in today's lesson, I will start with the history of solving equations. “

"Equations have a very high place in the history of mathematics, as early as more than 2,000 BC, people in the ancient Babylonian period already made unary equations, that is, ax+b=0. Needless to say, everyone should know that its root-finding formula is -b/a. But in the ancient Babylonian period, when there were only integers, how did the ancients understand b/a? This led to the introduction of the concept of fractions, in which fractions and integers are added together to be called rational numbers. “

Not only that, but the ancient Babylonians would also list a quadratic equation, that is, ax^2+bx+c=0, but the ancient Babylonians did not study this kind of equation thoroughly, and could only give some specific integer and fractional solutions. When the ancient Greek Pythagoreans arrived, they did not find the root solution of a quadratic equation, but they did find the solution of some specific equations. For example, when studying the Pythagorean theorem, they found that the diagonal length of a square with an edge length of 1 can be solved by a column equation. ”

Pang Xuelin got up and wrote on the blackboard: 1^2+1^2=x^2, x=√2.

So in order to solve the quadratic equation, irrational numbers were introduced. But have you noticed that there is no negative root here, the reason is very simple, the Europeans thought that negative numbers were meaningless, and it was not until the seventeenth century that Newton and Leibniz accepted the concept of negative numbers. In China, as early as the pre-Qin period BC, there was a concept of negative numbers, which is related to cultural traditions......"

"The root solution of the unary quadratic equation was first given by the Persian mathematician Khorazmi in the eighth century AD, but he only gave the positive root, and later his solution was transmitted to Europe, and after the introduction of the concept of negative numbers, it was slightly improved, which is the root solution of the unary quadratic equation as we know it now."

Pang Xuelin paused, picked up a bottle of mineral water on the podium and moistened his throat.

There was a buzzing discussion in the originally quiet classroom, and many students today are not from the mathematics department at all, they are all here to see the excitement and worship the god of learning, but they didn't expect that Pang Xuelin's lecture did not make everyone feel jerky, but there was a feeling of not hurrying and coming easily.

This amazed everyone.

After all, many big bulls are very academic, but they don't perform very well when they teach, either according to the script, or obscure.

On the contrary, Pang Xuelin broke down the history of mathematics and talked about it, giving everyone a refreshing feeling.

Pang Xuelin ignored the commotion below and continued:

"With the root solution, as long as the coefficients are substituted casually, it can be easily solved, so mathematicians began to look for the root solution of cubic equations and quadratic equations one after another."

"The root solution of the cubic equation was given by the Italian mathematicians Ferro and Tartaglia in the sixteenth century Renaissance, and Ferro gave the root solution of x^3+px=q, here you may think that this cubic equation does not have a general meaning, but if p and q are expressed as complex numbers, all cubic equations can be expressed in this form. But at that time, there was no concept of complex numbers, so Tartaglia, another Italian mathematician, gave the root solution of the general unary cubic equation ax^3+bx^2+cx+d=0, which is the so-called Cardano formula ......"

Pang Xuelin got up and wrote on the blackboard with a chalk brush, and spent more than half of the blackboard to express the Cardano formula.

"Have you noticed that in the Cardano formula, there is a problem of opening the root number to -3, but there are no complex numbers at that time, so people began to be interested in the problem of opening the root number of negative values, which led to the later complex number field. To some extent, in order to solve a cubic equation, the concept of complex numbers was introduced. A few years after Cardano's formula came out, one of Cardano's students, Ferrari, gave a formula for solving a quadratic equation. At this point, the root solution of the one, two, three, and fourth order equations has appeared. ”

So people thought that the formula for finding the root of the unary quintat equation was not far away, but they did not expect that in the next hundreds of years, people would not find the answer. So everyone began to find a way to simplify the problem, first to prove whether the unary quintessential equation has a root. Gauss proved that for any non-zero equation of non-zero unary n complex coefficients, there are exactly n complex roots. This is the fundamental theorem of algebra, and even if the discriminant formula of a quadratic equation is less than zero, it has two complex roots. Then the fifth-order equation should have five roots. “

Since there is a root, there should be a root solution, so people continue to search, and this problem was solved by the genius Norwegian mathematician Nils Abel. If you don't know who Abel is, you should have heard of the Abel Prize, one of the highest awards in mathematics, named after him. ”

Abel did not give the root solution of the fifth order equation, but he proved that there is no root solution for the fifth order equation. This is very powerful, in mathematics, it is often much more difficult to prove that something does not exist than to prove that it exists. ”

"This Abel is a character we are going to focus on today. Abel was born in Norway in 1802, and at the age of 17 he wrote a paper on his discovery of the root solution of a fifth-order equation. Later, he found that there were several errors in this paper, so he studied it diligently, continued to revise it, and four years later, he came to a new conclusion: there is no root solution for unary fifths. He also introduced a theorem called Abel Ruffini's theorem, but because this paper was too advanced for professional mathematicians at the time to understand, it did not attract attention at first. ”

Abel also showed Gauss this paper, and Gauss thought it was just a 21-year-old kid's vexatious trouble, which meant that it was difficult for Abel to get his paper published. Fortunately, Abel also has a friend named Clerler, he founded a mathematical journal, so Abel published this paper on this, in the next few years, Abel has made contributions in many fields, but mainstream mathematicians are not very acceptable, and he once sent his paper to the famous mathematician Cauchy, but Cauchy was even more cold, and he didn't even read it. Abel died young at the age of 27, and it was not until after his death that people discovered that his essays were classics. ”

"Abel's method of proving that there is no fundamental solution to a fifth-order equation is actually the group theory that we will learn in abstract algebra, but he did not put it forward systematically, but only used the idea of group theory to solve this problem. As a result, within a few years, another genius appeared, that is, Galois, who used the same idea of group theory to derive that there is no root solution for fifth-order equations, and he also gave in which cases there is a root solution for any higher order equation, and when there is no root solution. Moreover, Galois also proposed the concept of groups for the first time, creating a precedent for modern algebra. ”

"In my personal opinion, Galois's achievements are enough to rank in the top three in the history of mathematics. Galois was born in France in 1811 and, like Abel, thought he had discovered the root solution of a unary fifth order at the age of 16, but later proved that there was no root solution for a fifth order equation. At the age of nineteen, Galois threw himself into the French Revolution, was arrested and imprisoned at the age of 20, and after being released from prison at the age of 21, he was killed in a duel. But the paper was never accepted by the mathematical community until 1843, eleven years after Galois's death, the mathematician Liu Weil discovered the paper and published it, causing a huge sensation. The Galois theory ended the history of equation solving in mathematics for more than 4,000 years, and also opened the beginning of group theory. “

The Galois group theory also gives a general discriminant method for judging whether geometric figures can be drawn with a straightedge and a compass, which satisfactorily solves the problem that any angle or double cube of thirds is unsolvable. Most importantly, group theory has opened up a completely new field of research, replacing calculation with structural research, changing the way of thinking from focusing on computational research to thinking about using structural concepts, and classifying mathematical operations, so that group theory has rapidly developed into a new branch of mathematics, which has had a great impact on the formation and development of modern algebra. At the same time, this theory has had a great impact on the development of physics, chemistry, and even the emergence and development of structuralist philosophy in the twentieth century. ”