Volume 1: Student Life Chapter 169 Algebra and Number Theory
Chen Zhou glanced at his watch. The pace was good and there was still some time to spare.
He spent a total of 25 minutes of the 30-minute discussion time, reading through the four test papers of the other students and getting a good idea of the results.
Based on his understanding of each person, as long as they listened carefully to his instructions, there would be no big problem.
Things like exams sometimes depend on your performance on the spot.
Watching Yang Yiyi and the other three return to their seats, Chen Zhou began to look at his test paper.
When the time approached 9:30, the invigilator reminded everyone that no more discussion was allowed and each team member quickly returned to their seats and prepared to answer the questions.
The group written test started on time at 9:30 in the morning.
Following the command to start answering the questions, Chen Zhou began to write.
Start with question 1 in order.
Although the requirement was to choose 5 out of 6 questions, Chen Zhou didn't take this requirement to heart after reading through all 6 questions.
Chen Zhou planned to do 5 questions in order and then end.
Because he felt that these 6 questions were similar, the only difference was that some had one question mark, some had two or three question marks.
That is, the only difference is writing more formulas.
The first question of the "Algebra and Number Theory" test paper is an advanced algebra problem about Euclidean space.
【Let V = R^n be a Euclidean space, and g be an orthogonal matrix acting on V. When a∈V, there exists an inverse function Sa(x) represented by Sa: =x-[2(x,a)/(a,a)]a, Ax∈V. 】
The topic is not long, but it contains all the useful information.
After Chen Zhou read the questions again, he turned to question (1.1) without pausing.
【If a=（g－1）b≠0, please prove that ker（Sag－1）=ker（g－1）⊕Rb. 】
After reading the question, Chen Zhou started writing down his answer without pause.
Starting from the relationship between orthogonal matrices and Euclidean space, Chen Zhou's ideas are extremely clear and his writing is even more steady.
[…Since Sa(x):=x-[2(x, a)/(a, a)]a, Ax∈V…]
【……So ker(Sag-1)=ker(g-1)⊕Rb, proved. 】
Solve a small question and get 10 points.
Chen Zhou then went on to prove question (1.2).
Question 1 has two questions, each question is worth 10 points, for a total of 20 points.
The scores of the 6 questions are exactly the same, all of which are 20 points.
Choose 5 questions out of 100 points.
After Chen Zhou finished answering the two questions of Question 1, he started to answer Question 2.
This question has 4 small questions, each question is 5 points, the score is very evenly distributed.
The question type is still algebraic.
However, unlike question 1, this question examines the content of the exchange group.
The stem of this question is longer than the other questions.
But how should I describe the difficulty of the questions? Chen Zhou feels that the longer the question is and the more information it contains, the simpler it tends to be.
Moreover, the four questions in the title are internally connected and progressive, providing good help for answering the whole question.
Question 3.
After reading the questions, Chen Zhou smiled slightly. Finally, it came to the problem of algebraic integers.
Algebraic number theory is the study of algebraic integers and is an important branch of the number theory family.
Chen Zhou still remembers that when he was a freshman, the first textbook he read was "Basic Number Theory".
Therefore, it is inevitable that he feels some familiarity.
Moreover, the Goldbach conjecture, the pearl of number theory, has a special relationship with China.
This question is very short.
[Let ζ be a root of unity that satisfies the equation ζ = 1 + Nη (for integers N ≥ 3 and algebraic integers η). Prove that ζ = 1. ]
Chen Zhou picked up the pen, thought for a moment, and began to answer the questions.
Soon, question 3 was done.
Next come questions 4 and 5.
Question 4 is about finite groups.
Question 5 is about complex matrices and conjugate transformations.
Chen Zhou's thoughts were answered without any obstruction.
After he had fully solved these questions, he felt the same as when he first skimmed through them.
These problems may be more or less difficult for other people, but for him who has completed all undergraduate courses in the Department of Mathematics, they are not an obstacle at all.
In fact, just like he did during his freshman midterm exam, he didn't even use any scratch paper.
The answer to each question and the steps to each question appeared clearly in his mind when he finished reading the questions .
He simply changed the answers and steps from being calculated in his mind to actually writing them down on the test paper.
After taking a look at the test paper filled with math equations, Chen Zhou calmly stood up and handed in the test paper ahead of time.
Everyone in the examination room instantly focused their attention on Chen Zhou and watched him walk out of classroom 1308.
Chen Zhou also calmly accepted the gazes of these people. He was the first person to hand in his paper based on his own strength.
However, what Chen Zhou didn't know was that he was not only the first person to hand in his paper in the examination room 1308.
He was also the first person to hand in his paper at the Morningside Mathematics Center.
In all places in China where Qiu Sai team competition examinations are being conducted, whether it is Yanjing, Shanghai, Anhui Province, Jiangsu Province, Taiwan Province, or other places.
Chen Zhou was always the first one to hand in his paper.
After walking out of classroom 1308, Chen Zhou was not in a hurry to return to Yenching University.
After all, even if she didn't wait for Zhao Qiqi and the other two, Yang Yiyi still had to wait.
In the examination room, Yang Yiyi was looking at her test paper with half a question left. She pouted her lips, wrinkled her nose slightly, and muttered in her heart: "He is getting faster and faster now..."
Immediately, Yang Yiyi sped up her problem-solving speed, trying hard to catch up with Chen Zhou.
Unlike Yang Yiyi, there were many candidates in the examination room. They waited until Chen Zhou's figure completely disappeared at the door of classroom 1308 before they finally lowered their heads and continued to answer their questions.
However, these people's pace of answering questions was obviously affected by Chen Zhou.
They were really puzzled. How come Chen Zhou had already finished and handed in the paper when they had only done a little more than 3 questions?
These people are mainly students from Yenching University who have not taken the exam in the same examination room as Chen Zhou.
They had only heard of the legend of Chen Zhou but had never experienced it personally.
But I didn't expect that this first and only time I felt this way would make me feel so bad.
It took a long time for these people to calm down and regain their thinking on the test paper.
But before they could continue answering the questions, another person stood up and handed in his paper early.
These people inevitably looked up again.
At first glance, they felt that the person who handed in the paper early gave them an even worse impression than Chen Zhou.
Although she was the second one to hand in her paper.
But she is not a mathematics major!
That’s right, this person is Yang Yiyi.
After Yang Yiyi finished solving the last half of the question, she checked it twice more as Chen Zhou reminded her.
Avoid ignoring details of some steps and omit obvious problems.
After confirming that everything was correct, she didn't want to waste time, so she stood up and handed in the paper.
As Yang Yiyi's figure gradually approached the door of classroom 1308, it finally disappeared at the door of classroom 1308.
In the examination room of classroom 1308, most people's mentality was affected.
Originally, some people were able to withstand the impact of Chen Zhou submitting his paper early, but they were ultimately unable to withstand the impact of Yang Yiyi submitting her paper early again.
Just as Chen Zhou had thought before, things like exams sometimes depend on your performance on the spot.
There is no way. These people’s ability to withstand pressure is still a little lacking, which affects their on-the-spot performance.
Outside the examination room, Chen Zhou waved to Yang Yiyi as soon as he saw her coming out.
Yang Yiyi walked towards Chen Zhou in confusion.
Chen Zhou was chatting with one of the persons in charge of the Morningside Mathematics Center test site.
