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Chapter 169 Algebra and Number Theory

Chen Zhou glanced at his watch, the rhythm was good, and there was still some time left.

In the 30 minutes of discussion time, he spent a total of 25 minutes, reading the four sets of test papers of the others, and he knew what he was doing.

From what he knew of everyone, as long as they listened carefully to what he told them, it would be fine.

Things like exams sometimes depend on improvisation.

Watching Yang Yiyi and the others return to their respective seats and sit down, Chen Zhou began to read his test paper.

When the time was approaching 9:[-], the invigilator reminded that no further discussions were allowed, and the team members of each team hurried back to their seats and prepared to answer the questions.

At 9:[-] in the morning, the written test for the group test will start on time.

Following the command to start answering questions, Chen Zhou began to write.

Start with question 1 in order.

Although it was said that there were 6 questions and 5 answers, Chen Zhou didn't take this request to heart after browsing all the 6 questions.

Chen Zhou intends to do 5 questions in order and then finish.

Because he thinks that these 6 questions are similar, some have only one question mark, some have 2 or 3 question marks.

That is to say, write more formulas for the difference.

Question 1 of the "Algebra and Number Theory" test paper is about advanced algebra problems in Euclidean spaces.

[Assume that V=R^n is a Euclidean space, and g is an orthogonal matrix acting on V.When a∈V, there is an inverse function Sa(x) represented by Sa:=x-[2(x, a)/(a, a)]a, x∈V. ]
The topic is not long, but the useful information is complete.

After Chen Zhou read the question again, he looked at the 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 also started to answer without pausing.

Starting from the relationship between the orthogonal matrix and the Euclidean space, Chen Zhou's thinking is extremely clear, and his writing is more stable.

[…Since Sa(x):=x-[2(x, a)/(a, a)]a, x∈V……]

[...Therefore ker(Sag-1)=ker(g-1)⊕Rb, it is proved. 】

If you solve a small problem, you get 10 points.

Chen Zhou went on to prove the minor problem (1.2).

Question 1 consists of two small questions, each with 10 points, for a total of 20 points.

The scores of the 6 questions are exactly the same, all of which are 20-point questions.

Choose 5 questions out of 100 points.

After Chen Zhou solved all the two small questions of the first question, he began to answer the second question.

There are a total of 4 small questions in this question, each with 5 points, and the scores are very average.

The question type is still algebra.

However, unlike the first question, this question examines the content of the exchange group.

The stem of the title is longer than the other questions.

But the difficulty of the questions, how should I put it, Chen Zhou felt that the longer the questions, the more information the questions were, the simpler they would be.

Moreover, there are also internal connections among the 4 small questions of the title, and they are progressive layer by layer, which provides a good help for the answer to the whole question.

Question 3.

After reading the question, Chen Zhou smiled inwardly, and finally came to the problem of algebraic integers.

Algebraic number theory is the study of algebraic integers, and is an important branch of the family of number theory.

Chen Zhou still remembers the basic textbook "Basic Number Theory" that he first read when he was a freshman.

Therefore, he inevitably has some intimacy.

Moreover, Goldbach's conjecture, the jewel in number theory, has an extraordinary relationship with Huaguo.

This question is very short.

[Let ζ be the unit root satisfying the equation ζ=1+Nη (for integer N≥3 and algebraic integer η).Prove that ζ=1. 】

Chen Zhou picked up the pen, thought for a while, and began to answer the questions.

Soon, question 3 will be done.

Then there are questions 4 and 5.

Problem 4 is about finite groups.

Question 5 is about complex matrices and conjugate transformations.

Chen Zhou answered without the slightest hindrance.

After answering these questions comprehensively, he felt the same as when he first browsed the questions.

These questions are more or less difficult for others, but for him who has completed all the undergraduate courses of the Department of Mathematics, there is no obstacle at all.

Even, just like he used to do during the senior high school entrance examination, he didn't even use scratch paper.

The answer to each question and the steps of each question clearly appeared in his mind when he finished reading the questions.

He just changed these answers and steps from calculating in his mind to actually writing them down on the test paper.

After glancing at the paper full of calculation formulas, Chen Zhou stood up calmly and handed in the test paper in advance.

Everyone in the examination room focused their eyes on Chen Zhou for a moment, and watched him walk out of classroom 1308.

Chen Zhou also accepted the gazes of these people calmly. He was the first to hand in the test based on his strength.

However, what Chen Zhou didn't know was that he was not only the first person to hand in the examination paper in classroom 1308.

He was also the first person to hand in the exam at the Morningside Math Center.

In all the places in Huaguo where the team competition exams are being held, whether it is Yanjing, Shanghai, Anhui Province, Jiangsu Province, Taiwan Province, or other places.

Chen Zhou was the first to hand in the paper.

After walking out of classroom 1308, Chen Zhou was not in a hurry to return to Yenching University.

After all, even if they didn't wait for Zhao Qiqi and the three of them, Yang Yiyi still had to wait.

In the examination room, Yang Yiyi was looking at her paper with half of the questions left, pouted her mouth, wrinkled her nose slightly, and muttered in her heart: "He is getting faster and faster now..."

Immediately, Yang Yiyi accelerated her speed of solving the problem, trying to catch up with Chen Zhou's footsteps.

Unlike Yang Yiyi, there were quite a few 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 own questions.

It's just that these people were obviously affected by Chen Zhou's rhythm of answering the questions.

They were really puzzled, how could Chen Zhou have already completed and handed in the paper after he had only done 3 more questions?
These people are mainly students of Yenching University who have not taken the exam in the same exam room as Chen Zhou.

They have only heard the legend of Chen Zhou, but they have never experienced it personally.

But I didn't expect that this first and only feeling made me feel bad.

It took a lot of time for these talents to stabilize their emotions and regain their thoughts on the test papers.

But before they could continue to answer the questions, another person stood up and went to hand in the paper ahead of time.

These people inevitably looked up again.

At this glance, they only felt that the person who handed in the test earlier gave them more feelings than Chen Zhou.

Although she was the second to hand in the paper.

But she is a non-math student!

That's right, this person is Yang Yiyi.

After Yang Yiyi finished answering the remaining half of the questions, she followed Chen Zhou's reminder and checked it twice.

Avoid ignoring the details of some steps, and obvious omissions.

After confirming that it was correct, she didn't want to waste time, so she got up and handed in the paper.

As Yang Yiyi's figure gradually approached the classroom door of 1308, until it finally disappeared at the classroom door of 1308.

In the examination room of classroom 1308, the mentality of most people was affected.

Originally, some people had resisted the influence caused by Chen Zhou handing in the paper early alone, but in the end they couldn't bear the impact caused by Yang Yiyi's early handing in the paper again.

As Chen Zhou thought before, things like exams sometimes depend on improvisation.

No way, these people's ability to resist pressure is still a bit weak, which affects their on-the-spot performance.

Outside the examination room, as soon as Chen Zhou saw Yang Yiyi coming out, he waved to her.

Yang Yiyi walked towards Chen Zhou suspiciously.

Chen Zhou was chatting with one of the test center directors of the Morningside Math Center.

(End of this chapter)