If you remember, I once said that the mathematical tool used in RSA encryption is group theory, and Galois can be said to be the most important mathematician who created group theory.
Although he was called a mathematician, he actually died in a gun battle duel with others at the age of 2 1.
This bizarre life experience makes the birth of group theory more romantic.
Let me talk about his life first.
In his short life of 2 1 year, only the last five years can be regarded as studying mathematics. Before that, he was a child. Like all children, he needs to go to school, finish his studies, do his homework and lead a normal life. But his life is not peaceful, and it is not easy to be a happy student.
Seven years before his birth; Napoleon proclaimed himself emperor;
At the age of 3, Napoleon was ousted again;
At the age of 4, Napoleon came back;
At the age of five, Napoleon was finally wiped out and the Bourbon dynasty was restored again.
From then until his death, during the period of 15 years, the French people have been violently resisting the absolute monarchy of Bourbon dynasty.
/kloc-in 0/9, the July Revolution finally broke out, which overthrew the autocratic rule of France forever, and then France was transformed into a constitutional monarchy.
Before he was 1 1, his mother taught him to read and write. When he reached middle school age, his family specially sent him to a boarding school with militarized management. In fact, this kind of school has its advantages in turbulent times, that is, it can protect students, otherwise middle school students will easily be incited by public opinion and become cannon fodder in the streets.
At that time, Galois did well in school and should have graduated a year earlier, but the principal just thought he was too young to agree. So Galois's last year in this middle school is actually equivalent to rereading Grade Three for the second time.
He also began to learn mathematics from this year, not only because he had more leisure time, but also because he met a good teacher, M.Vernier.
What did junior high school students in France learn 200 years ago? In fact, it is similar to what we are learning now, such as solving equations, and the highest solution is binary once.
But galois has mastered it for a long time. Under the guidance of Wiener, he began to go further along the road of solving equations and began to read some works about the properties of equation solutions. These contents are the only core of his research as a mathematician.
However, at the age of 14, far from being a mathematician, he will continue to study in high school. After graduating from high school, he will go to college.
Galois learned a lot of modern mathematics very early, and was seriously biased from the age of 14, so he failed to enter the Paris Institute of Technology for the first time.
In France, this school is equivalent to Tsinghua University and China, so he retakes the exam for one year. This time, because of the teacher's strong recommendation, the college made an exception and only gave him an oral exam. However, Galois skipped many steps because the questions asked by the interviewer were too simple, which made the interviewer unable to understand, and the two sides immediately misunderstood. It is said that Galois was so angry that he threw a rag to clean the blackboard in the interviewer's face.
Naturally, he failed in the second exam, but he can't repeat it.
Because the school has a rule-those who have not been admitted for more than two times will never be admitted. In the end, he can only choose to take the normal college affiliated to this college.
It was in the preparation stage that he made his first valuable academic paper, which was about the analysis of positive integer roots of equations. This paper puts forward the concept of "group".
There is a great academic background here: since 1500s, European mathematics has returned to the peak level of ancient Greece. At that time, a hot question was how to solve the equation.
Before Galois was born, formulas for finding the roots of quadratic, cubic and quartic equations came out one after another, that is, as long as the coefficients from x 4 to x 0 are known, all solutions can be calculated by a general formula.
For example, X = [-b (b 2-4ac) (1/2)]/2a, which we require to recite in junior high school, is the formula for finding the root of quadratic equation. In fact, there are cubic equations and quartic equations respectively, but because the formulas are too long, we are not required to recite them.
But all mathematicians are stuck in the formula of finding the roots of fifth-order and higher-order equations, and Galois studied this problem that year.
His first breakthrough was to prove that the quintic equation does not necessarily have a formula for finding the root;
The second breakthrough is to analyze the characteristics of the equation and find the root formula.
The analytical tool is the concept of "group" invented by himself, which was later called "Galois Group".
What exactly is a "group"? In fact, a formal statement should start with the definition, namely:
Satisfying these four elements can form a group.
Although this is simple to say, it is difficult for people to appreciate the beauty of the group, and even make people feel that such a collection is nothing special. Now let's feel the charm of the group from another angle and see what the group can analyze.
This kind of thing is a tool to peel off the appearance of things and reach the essential attributes. For example, there are the following three groups of questions. Let's draw a cube. These three questions are:
You may think that these questions are weak, even primary school students know them, but recall that there are three groups of numbers in the three groups of questions just now, which are 6 and 4, 12 and 2, 8 and 3 respectively. Are all their products 24?
Do you think this is a coincidence that I deliberately pieced together? Actually, it is not. This 24 is the total possibility of placing this cube.
So, the next question comes again:
We can calculate this way: in the first position, we can choose any one of the four balls to put there, so the possibility is multiplied by 4; There are only three balls in the second position, so the possibility should be multiplied by 3; Third position, fourth position, and so on. Therefore, the total number of permutations is 4×3×2× 1=24.
Then the last question is: the arrangement of cubes and balls is 24. Is there any internal connection between these two 24 s?
Actually, there is. Their correlation is that their mathematical structures are the same. These contents can be analyzed by group theory.
After feeling the charm of group theory, let's take a look at Galois.
The fate of his thesis at that time was extremely tragic.
After the manuscript was submitted to the French Academy of Sciences, Cauchy, a promising young mathematician at that time, came to review the manuscript. However, Cauchy had too many things to do and it took seven months to reply. Cauchy said that he plans to introduce Galois's views on this article at the next regular meeting of the Academy of Sciences. But on the day of the meeting, Cauchy spent all her speech time introducing her paper, and Galois didn't mention a word.
In these seven months, Galois did not wait, but optimized and re-optimized the paper and submitted it to the French Academy of Sciences again. The critic is even more famous this time, Fourier. As a result, the paper was taken away by Fourier for three months without any reply, and people didn't know that the old man had passed away until they asked.
I waited for a whole year, but there was no result.
In fact, the ideas contained in this article are important enough, but because it was written by a young man of 18 years old, no one paid attention to it, so this grumpy young man lost his last chance to take the academic road.
From the previous college entrance examination oral test, we felt that the young man threw the rag at the examiner's face and lost his temper. Indeed, his energy was spent not only on mathematics, but also on disgusting things.
When he was in middle school, he often took the lead in challenging the headmaster, such as asking him to allow them to carry guns for military training on campus, and asking him to lift the ban that they can only go out once a month.
After the school warned him, he published a long article attacking the principal in the school magazine, and finally he was really fired. He joined the National Guard immediately after being expelled, and was later detained for several days for riots.
Just like the time when he was expelled from school, a little warning would not restrain him, but would ignite the fighting spirit of this grumpy young man. As a result, shortly after his release from detention, he led a team to participate in an armed parade celebrating the 40th anniversary of the capture of the Bastille. The slogan of the parade was to guillotine the then emperor Louis Philips.
This time, he was arrested again. The punishment this time is not detention for a few days, but imprisonment for six months. Although he was imprisoned, he often preached the idea of burying the king alive, so his sentence was increased to 15 months after 6 months.
In fact, he was released after only two months in prison, and the rest were executed outside the prison. The reason is not because he has access, but because there was a cholera epidemic in France at that time, and this cholera was the most serious in the history of human medicine. For the sake of safety, the prisoner can only evacuate. Galois was sent to a rehabilitation center dozens of kilometers away and executed outside the prison.
In the rehabilitation home, Galois met his first love Stephanie. This girl is the daughter of the owner of the rehabilitation home.
From then on, Galois will go to death step by step.
The girl's attitude towards him is indifferent, sometimes cold and sometimes hot, and Galois is sometimes disheartened and sometimes enthusiastic. Stephanie's name often appears repeatedly in his final draft of mathematics. In fact, his first love is very complicated. He may be a spy and bear a grudge against the party supported by Galois.
Later historians analyzed that Galois himself should have known this dilemma at that time, but he was trapped by love and could not get rid of it. 1832 On May 28th, he received a letter of challenge, inviting Galois to shoot with him in the tone of rival in love.
Realizing that his time was running out, Galois seized the days of May 28th, 29th and 30th to perfect his content about group theory. In the margin of the reserved draft, you can often see the sentence "I don't have enough time"
On the evening of 30th, he wrote three more suicide notes, two of which were addressed to himself * * * and party member, and 1 to Qunlun, and left them to his good friend Auguste.
In the gunfight the next morning, Galois lost to a professional soldier, was shot three times in the abdomen, and died one day after being taken to the hospital.
Galois's life is over. His friend August is very responsible. He spent several years sorting out Galois's manuscripts and then sent them to joseph liouville, a famous French mathematician at that time.
Joseph liouville realized the value of this material, sorted it out and standardized it, and published The Thought of Group Theory in 1846 to replace Galois.
After 10 years, group theory developed rapidly. At that time, in most universities in France and Germany, mathematics majors had begun to teach Galois Group Theory.
At this time, the political situation in France has also been initially stable. But that year, Galois had been dead for 24 years.
There are many things worth thinking about in Galois's story, but from an academic point of view, we can think about:
If Galois has a stable personality, he will be able to enter the comprehensive technical college smoothly, get a degree, and get a teacher-student relationship in mathematics. If this is the case, the papers written in the future will be recognized by the academic circles from format to expression.
But he didn't have such a personality and didn't enter the academic circle, so his tragic fate was actually determined by his personality and times.
In the next part, genius doctor, a gifted physicist and a gifted linguist deciphered the ancient Egyptian script-Thomas Young.