A genius? I just love studying.

Chapter 67 This is not the way to school (please read)

All constructible numbers form a field K∈R, and if a>0 is a constructible number, then the square root a is also constructible.

Since O and A are two points given in advance, 1 is constructible, so all rational numbers Q are constructible, so K is a field extension of Q.

Therefore, the method for determining constructible numbers is as follows:
r∈R is constructible if and only if there exists a domain extension tower, Q=k0∈k1∈…∈kn, such that r∈kn, and the adjacent domain extension index [ki+1, ki]≤2, so if r∈K, then r is an algebraic number of Q, and [Q[r]:Q]=2^m.

After writing this, Chen Hui's thoughts became completely clear, and he then looked at the first question that needed to be proved.

1. Since the trisection of π/3 can be constructed if and only if cos(π/9) = α is a constructible real number, consider implementing π/9 as the interior angle of a triangle whose three sides are all constructible.
根据三倍角公式cos3θ=4cos^3θ-3，所以有4α^3-3α=1/2，

因此α满足多项式方程f(x)=8x^3-6x-1=0。

According to the Eisenstein criterion, f(x) is an irreducible polynomial in Q[x], so [Q[α]:Q]=3>2.
Therefore, it is not a constructible real number, so it is impossible to use a ruler and compass to trisect an angle of π/3.

All in one go!
It only took Chen Hui a dozen minutes to prove the first question.

Then he looked at the second question.

With the previous thinking, the following questions are just a matter of copying. Although the fourth question requires some more complicated processing, it is not too difficult.

Zhang Anguo stood there blankly, not knowing what to say.

Chen Hui was actually able to write a complete proof based on just an abstract algebra textbook. What kind of monster is this?
This guy's talent in algebra and number theory can only be compared with others in math textbooks!
He is now becoming more and more confident in Chen Hui. He believes that even if Chen Hui cannot win the gold medal, he still has a good chance of winning the silver and bronze medals.

Time passed slowly. Liang Peixuan, who had finished a geometry proof, stretched and looked back with some curiosity.

Zhang Anguo's face, which was so shocked that he had lost control of his expression, made him even more curious. He lowered his head slightly and looked at the draft paper where Chen Hui was writing furiously.

"?"

He actually recognized all the words on the draft paper.

"No, that's not right. Why is there a double image of R?"

"Are there brackets in math formulas?"

"What is a domain, what is a tower, what is constructible"

Countless question marks popped up in Liang Peixuan's mind.

What exactly is Chen Hui writing?
How come he can't understand it at all?

This, this is mathematics?

Looking back at the geometry proof he had just completed, Liang Peixuan suddenly felt a little bored for some reason.

If he had only felt that he was getting further and further away from Chen Hui before, now he felt that he and Chen Hui were no longer from the same world.

Chen Hui breathed a sigh of relief, finished writing the last symbol, and put down the pen.

I also feel a sense of accomplishment.

It's a wonderful feeling to use mathematical tools, especially to come up with a great entry point, use a clever conversion method... and finally solve a difficult problem!

In this way, Chen Hui solved several problems that had stumped ancient Greek mathematicians for their entire lives in just half an hour!

Jingle Bell……

Almost at the same moment when Chen Hui put down his pen, the bell on campus rang, urging the students to leave.

The ringing of the bell woke Zhang Anguo from his shock. He licked his dry lips and swallowed his saliva to moisten his equally dry throat. "Next, you can start studying analytic geometry and learn about differential geometry and Riemannian geometry."

"Topology is also a very important mathematical tool. You must know point set topology and algebraic topology, and then you should learn about Lie groups and Lie algebras. This is a very popular field, and the competition will likely involve relevant questions." "Also, you need to learn how to use LaTeX now. When the questions are answered online, many formulas will need to be edited using LaTeX. Starting tomorrow, I will bring a notebook and teach you for an hour every day. You should familiarize yourself with it first, and you should be able to use it proficiently in two or three days."

As he spoke, Zhang Anguo became more confident. Fortunately, he was born more than 20 years earlier than Chen Hui. Even if Chen Hui was a genius, there were still many things he didn't know, and as a teacher, he still had things to teach him.

Of course, he didn't know how long his inventory could last, a month or a year?
"Ah."

Chen Hui nodded.

Zhang Anguo had already given him all the textbooks in advance, and of course he also knew about LaTex.

After getting a mobile phone, he would also spend some time searching for information related to the Barbaria Mathematical Competition. Now he is not completely ignorant of the Barbaria Mathematical Competition.

……

June 3rd, 6:40 in the morning.

While many of her classmates were still fast asleep, Cai Jingjing had already left home and walked towards school with her schoolbag on her back.

Although he did not regard the students of Rongcheng No. 2 Middle School as his rivals, the news that kept coming from that outstanding alumnus still gave him a strong sense of crisis. He felt that he had to work harder!

Although the popularity of that genius has declined recently, he still maintains this habit. He believes that as long as he works harder, he will never be worse than that person!
Suddenly, he saw two figures with schoolbags on their backs in front of him, also walking towards Rongcheng No. 2 Middle School.

There is actually someone who goes to school earlier than him?

Cai Jingjing subconsciously took a few quick steps forward.

Soon, he recognized one of the two people in front of him. It was Chen Hui, the talented student who was recently in the limelight at Rongcheng No. 2 Middle School.

He had never had any contact with Chen Hui, but news about Chen Hui had been everywhere recently, and he had seen Chen Hui's photos many times.

"Slip away!"

“Adding filetype:txt when searching for novels really works!”

"There are basically no ads in the search results. You are a genius!"

"I'll never have to worry about downloading e-books again!"

Li Hai's joy was beyond words, and his voice, which he did not try to lower, naturally reached Cai Jingjing's ears.

"This is a simple search technique. Adding a - or plus sign can effectively filter information, turn off the system's default recommendations, and enable advanced search mode. The quality of the word actually only determines the subsequent accuracy. Replacing most words will work, and even any keyword can reduce the weight of ad recommendations."

Chen Hui's calm voice rang out, "Not just Baidu, the overall logic of most search engines is that default search = default priority advertising, adding any keywords = enabling advanced search = default reduction of advertising recommendations.

There are many other logical symbols that are available on major search engines. The Xiamen University Library has a related training courseware. If you are interested, you can search for it.

"..."

"You've only been using computers for a few days, and you're already better at it than I am!"

Li Hai couldn't help but roll his eyes. He had been on the Internet for so long, but he had never heard of such a thing?
Cai Jingjing didn't understand what the two were talking about, but as he followed them, he suddenly frowned.

This doesn't seem to be the way to school, right?

From now on, I will post two chapters at a time so that everyone can read enough at once.

Today is Tuesday and I am still a little stuck in reading. The second chapter will be around 11pm.

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(End of this chapter)