# 无招胜有招 – 抽象数学Abstract Algebra 的威力

Évariste Galois (1811-1832) : 21岁的天才，一人奠立Abstract Algebra 的公理。4位当年世界数学第一流的大宗师 (高斯Gauss, Fourier, Cauchy, Poisson) 都看不懂。

Galois 被 Ecole Normale Superieure “ENS” (培养世界最多Fields Medalists ) 踢出校门，前2年（16＆17岁）他参加 每年一次的＂高考＂Concours （法国’科举’ ）落榜 2次, 被(法国排名第一的) Ecole Polytechnique (外号”X” )拒绝入学 。他死后14年，”X” 的Louisville 教授发现他的论文 “Abstract Algebra / Group Theory” ， 公布于世。200年后 ENS 校长正式给Galois一个＂迟来的道歉＂。

https://tomcircle.wordpress.com/2020/07/24/%e6%97%a0%e6%8b%9b%e8%83%9c%e6%9c%89%e6%8b%9b-%e6%8a%bd%e8%b1%a1%e6%95%b0%e5%ad%a6abstract-algebra-%e7%9a%84%e5%a8%81%e5%8a%9b/

[NOTE]

Victor Hugo 写的《Les Miserables》歌剧里，革命党的学生领袖之一就是Galois (爬上马车) , 和腐败的＂保皇党＂对抗。Galois 被捕入狱半年，有空闲时间把 Group Theory 的草稿重新修改整理。出狱后不久就和人枪斗, 被情敌 (政敌) 杀害。死前一夜，还赶工整理群论文章， 纸张边涂＂je n’ai pas le temps＂(我没有时间了…)

[Reference]

# 史上最悲惨的数学家是谁？为什么不能三等分任意角？【尺规作图2/2】

Key Points:

1. Three Ancient Greek Problems : a) Trisect an angle“, b) Doubling the Cube, c) Square a Circle
2. Field Theory
3. Abel
4. Galois
5. Proved by Wantzel 2 out of 3 Greek Problems: 1a) & 1b)

Note :1837 Pierre Wantzel proved 2 out of the 3 ancient Greek Problems “Trisect an angle” & Double a Cube, only 5 yrs after Galois had published his “Group & Field” Theories before death at 21.

Wantzel was the only person in French history who scored 1st in both Concours (法国抄袭中国的数学“科举“) of Ecole Normale & Ecole Polytechnique.
His life regret was switching from his talent in Math to becoming a mediocre Civil Engineer.

https://en.m.wikipedia.org/wiki/Pierre_Wantzel

# “Poisson Distribution Intuition (and derivation)”

“Poisson Distribution Intuition (and derivation)” by Aerin Kim 🙏

https://link.medium.com/bho611jxSX

Poisson was a 19CE French mathematician, or Statistician, who invented the Poisson Distribution.

Note: “Applied” Math vs “Pure” (aka Modern / Abstract) Math

Poisson, being an Applied Math expert in the French Science Academy, did not understand the “abstract” Math paper submitted to him by the 19-year-old Evariste Galois in “Group Theory” for the “Insolvability of Quintic Polynomial Equations“. Poisson rejected the paper commenting it was too vague, destroying the final hope of Galois to make his invention the “Modern Math” known to the world. The dejected poor Galois died at 21 after a duel during the chaotic era of the French Revolution.

# Louis-Le-Grand, un lycée d’élite 法国(巴黎)精英学校: 路易大帝高级中学

Lycée Louis-Le-Grand, founded since 1563, is the best high school (lycée, 高中 1~3) for Math in France – if not in the world – it produced many world-class mathematicians, among them “The Father of Modern Math” in 19th century the genius Evariste Galois, Charles Hermite, the 20th CE PolyMath Henri Poincaré, (See also: Unknown Math Teacher produced two World’s Math Grand Master Students ), Molière, Romaine Rolland (罗曼.罗兰), Jean-Paul Satre, Victor Hugo, 3 French Presidents, etc.

Its Baccalauréat (A-level) result is outstanding – 100% passed with 77% scoring distinctions. Each year 1/4 of Ecole Polytechnique (*) (France Top Engineering Grande Ecole ) students come from here.

More surprisingly, the “Seconde” (Secondary 4 ~ 中国/法国 “高一”) students learn Chinese Math since 6ème (Primary 6).

Note : Below is the little girl Heloïse (on blackboard in Chinese Math Class) whose admission application letter to the high school :

Translation – I practise Chinese since 6ème (Primary 6), 5 hours a week. I know that your school teaches 1 hour in Chinese Math, which very much interests me because Chinese and Mathematics are actually the 2 subjects I like most.

Interviewer asked Heloïse :

Q: Why do you learn Chinese?

A: It is to prepare (myself) for working in China in the future, to immerse now in the language of environment. Anyway, the Chinese mode of operation is so different from ours.

Note : Louis Le Grand (= Louis 14th). He sent in 1687 AD the Jesuits (天主教的一支: 耶稣会传教士) as the “French King’s Mathematicians”(eg. Bouvet 白晋) to teach the 26-year-old Chinese Emperor (康熙) KangXi in Euclidean Geometry, etc.

Note (*): 5 Singaporeans (out of 300+ French Scholarship students) had entered Ecole Polytechnique through Classes Préparatoires / Concours aux Grandes Ecoles in native French language since 1980 to 2011. It is possible one day some of these elite French boys and girls could enter China top universities via “Gaokao” (高考 ~ “Concours”) in native Chinese language.

# Quora: Galois Field Automorphism for 15/16 year-old kids

3 common Fields: $\mathbb{R, Q, C}$ with 4 operations : {+ – × ÷}

Automorphism = “self”  isomorphism (Analogy:  look into mirror of yourself,  image is you <=> Automorphism of yourself).

The trivial Field Automorphism of : $\mathbb{R, Q}$ is none other than Identity Automorphism (mirror image of itself).

Best example for Field Automorphism: $\mathbb{C}$ and its conjugate. (a+ib) conjugate with (a-ib)

Field automorphisms using terms a 15/16/ year old would understand? by David Joyce

What interesting results are there regarding automorphisms of fields? by Henning Breede

# Group Theory in Rubik’s Cube & Music

Group is the “mathematical language” of Symmetry — the beauty which pleases human’s eyes and animal’s eyes: We are attracted by a symmetrical face (五官端正), bees are attracted by a symmetrical flower…

Group was discovered by a 19-year-old French genius Evariste Galois (sound: \ga-lua, 1811-1832), who was attempting to explain why any quintic equations (polynomial with degree 5 or more) could not have a solution formula (like quadratic, cubic, equations) using +, -, ×, /, nth root radicals. This 300-year problem since 16th century defeated even then the world’s greatest Mathematician Gauss. Before a fatal duel which killed him, Galois wrote down his discovery of ‘Group Theory’ which was understood only 14 years later by Professor Louisville of the Ecole Polytechnique — ironically the same university which failed Galois twice in admission exams (Concours, aka French imitation of Chinese 科举).

Group has 4 properties: “CAN I ?”

C: Closure
A: Associative
N: Neutral element (or Identity)
I: Inverse

Rubik’s cube is group. It is a fun game.

Music is group. It is pleasing to ears if the music is nice, or “symmetric”.

# 费马大定理 Fermat’s Last Theorem

1977秋 ~1979秋 笔者在法国-图卢斯(Toulouse, Southern France, Airbus 产地)费马学院 (College Fermat, aka Lycée Pierre de Fermat: Classe Préparatoire, 178th Batch)读两年的大学近代数学 (Mathématiques Supérieures et Spéciales), 尝过一生读书的”地狱”生活, 严谨(Mathematical Rigor)的思考训练, 像地鼠般(法国人戏称taupe)不见天日, 废寝忘食的煎熬。 当年对数学的恐惧, 终生牢牢铭记在心; 30年后”由惧转爱”, 数学竟然成为半退休后的业余嗜好, 享受数学的美 — 也是造物者宇宙天地的美!

FLT 350年数学长征英雄人物:
1. Fermat (费马 1601@ Toulouse, France)
2. Galois (伽罗瓦): Group Theory (群论)
3. Gauss (高斯)
4. Cauchy (柯西) Lamé (拉梅) Kummer (库马)
5. Solphie Germain
6. Euler (欧拉)
7. Taniyama (谷山丰), Shimura (志村五郎)

“数风流人物, 还看今朝”集大成者 :
8. Andrew Wiles (怀尔斯) 证明 (1994 -1995)”盒外思路” (Think Out of The Box): The Great Moment of 1994 Proof (YouTube)

$\boxed {(1) = (2) = (3) }$
(1). Elliptic Curve (椭圆曲线)
(2). Modular Form (模形式)
(3). Fermat’s Last Theorem (费马大定理)

# Great Math Popular Books

Evariste Galois‘s genius is he built a “bridge” between Field (域/体) and Group (群) – both new concepts invented by him. The “bridge” is called Galois Group, or by Emile Artin the “Group Automorphism”. He transferred the difficult problem of solving complicated 4 -ops (+-×/) Field (coefficients) to the single-op (permutation of roots) Group.

Galois Group is the ultimate TRUTH of all Math — Fermat’s Last Theorem, and any advanced Math, will use Galois Group or Field, to solve. Prof C.N. Yang 杨振宁 Nobel-Prize Physics discovery was based on Group Theory.

Evariste Galois was a French Math genius, died at 21 in a duel during French Revolution. He is the ‘Father’ of Modern Algebra. Failed 2 years in Ecole Polytechnique CONCOURS Entrance Exams, then kicked out by Ecole Normale Supérieure, his Math was not understood by all the 19th century World’s greatest Math Masters : Gauss, Fourier, Poisson, Cauchy…

“Galois Theory” — the ultimate Math “葵花宝典” (a.k.a. “Kongfu Bible“) — is only taught in the Math Honors Undergraduate or Masters degree Course.

— “高中会教这种困难的数学吗 ?”
— “…我觉得比起給高中老师教, 不如自己好好学吧。”
— ” 重要的是自己学习。”

http://m.ruten.com.tw/goods/show.php?g=21437146332387

http://www.nh.com.tw/nh_bookView.jsp?cat_c=01&stk_c=9789866097010

https://tomcircle.wordpress.com/2014/03/21/math-girls-manga/

# French Concours & 科举 (Chinese Imperial Exams)

French Concours (Entrance Exams for Grandes Écoles) was influenced by Chinese Imperial Exams (科举\ko-gu in ancient Chinese, today in Hokkien dialect) from 7th century (隋朝) till 1910 (清末).  The French Jesuits priests (天主教耶稣教会) in China during the 16th -18th centuries ‘imported’ them to France, and Napoléon adopted it for the newly established Grande École Concours (Entrance Exams), namely, “École Polytechnique” (a.k.a. X).

The “Bachelier” (or Baccalauréat from Latin-Arabic origin) is the Xiu-cai (秀才), only with this qualification can a person teach school kids.

With Licencié (Ju-ren 举人) a qualification to teach higher education.

Concours was admired in France as meritocratic and fair social system for poor peasants’ children to climb up the upper social strata — ” Just study hard to be the top Concours students”! As the old Chinese saying: “十年寒窗无人问, 一举成名天下知” (Unknown as a poor student in 10 years, overnight fame in whole China once top in Concours). Today,  even in France, the top Concours student in École Polytechnique has the honor to carry the Ensign (flag) and be the first person  to march-past at Champs-Elysées in the National Day Parade.

Concours has its drawback which, albeit having produced top scholars and mandarins, also created a different class of elites to oppress the people. It is blamed for rapidly bringing down the Chinese Civilization post-Industrial Age in the last 200 years. 5 years before the 1911 Revolution, the 2nd last Emperor (光绪) abolished the 1,300- year-old Concours but was too late. Chinese people overthrew the young boy Emperor Puyi (溥仪) to become a Republic from 1911.

A strange phenomenon in the1,300-year Concours in which only few of the thousands top scorers — especially the top 3 : 状元, 榜眼, 探花 e.g. (唐)王维, (北宋)苏东坡, 奸相(南宋)秦桧，贪污内阁首輔(明)严嵩… — left their names known in history, while those who failed the Concours were ‘eternally’ famous in Literatures (the top poets LiBai 李白 and DuFu 杜甫)， Great writers (吴承恩, 曹雪芹, 蒲松龄, 罗贯中, 施耐庵), Medicine (《本草綱目》李时珍, 发明”银翹散”的吳鞠通), Taipeng Revolution leader (洪秀全)….

Same for France, not many top Concours students in X are as famous in history (except Henri Poincaré) as Evariste Galois who failed tragically in 2 consecutive years.

The French “grandiose ” in Science – led by Pascal, Fermat, Descartes, Fourier, Laplace, Galois, etc. — has been declining after the 19th century, relative to the USA and UK,  the Concours system could be the “culprit” to blame, because it has produced  a new class of French “Mandarins”  who lead France now in both private and government sectors. This Concours system opens door to the rich and their children, for the key to the door lies in the Prépas (Classes Préparatoires, 2-year post-high school preparatory classes for grandes écoles like X), where the best Prépas are mostly in Paris and big cities (Lyon, Toulouse…), admit only the top Baccalauréat (A-level) students. It is impossible for poor provinces to have good Prépas, let alone compete in Concours for the grandes écoles. The new elites are not necessary the best French talents, but are the privilegés of the Concours system who are now made leaders of the country.

Note: Similar education & social problem in Japan, the new Japanese ‘mandarins’ produced by the competitive University Entrance Exams (Todai 东大) are responsible for the Japanese post-Bubble depression for 3 decades till now.

These ‘Mandarins’ (官僚) of the past and modern days (Chinese, French, Korean “Yangban 양반 両班 “, Japanese) are made of the same ‘mould’ who think likewise in problem solving, protect their priviledged social class for themselves and their children, form a ‘club mafia’ to recruit and promote within their alumni, all at the expense of meritocracy and well-being of the corporations or government agencies. The victim organisation would not take long to rot at the roots, it is a matter of time to collapse by a sudden storm overnight — as seen by the demise of the Chinese Qing dynasty, the Korean Joseon dynasty (朝鲜李氏王朝), and the malaise of present French and Japanese economies.

# Our Daily Story #4: Niels Henrik Abel, a poor Math genius

Abel and Galois (Story #3) had many things in common: both worked on the Quintic equation (of degree 5). Abel first proved there was NO radical solution; Galois, who was 9 years younger, went one step further to explain WHY no solution (with Group theory).

Both were young Math genius not recognized by the world of mathematics. Their fates were ruined by the same French mathematician Augustin Louis Cauchy, who was infamous of selfishly ignore other’s achievement but his own, hid the two’s Math papers from the recognition of the French Academy of Sciences.

Both died young: Abel at 26,  Galois 20.
Abel was poor and weak in health. His dream job of professorship came 2 days (too late) after his death.

Ironically, today the top Math award – in monetary term US\$ 1 million – for the world’s top mathematician (regardless of age) is named after this extremely poor mathematician – the Abel Prize.

http://scienceworld.wolfram.com/biography/Abel.html

(Go to YouTube – search “Niels Henrik Abel” – to read the English sub-title of this Norwegian video.)

# Our Daily Story #3: The Math Genius Who Failed Math Exams Twice

To prove the FLT, Prof Andrews Wiles used all the math tools developed from the past centuries till today. One of the key tool is the Galois Group,  invented by a 19-year-old French boy in 19th century, Evariste Galois. His story is a tragedy – thanks to the 2 ‘incompetent’ examiners of the Ecole Polytechnique (a.k.a. “X”), the Math genius failed in the Concours (Entrance Exams) not only once, but twice in consecutive years.
Rejected by universities and the ugly French politics and academic world, Galois suffered set back one after another, finally ended his life in a ‘meaningless’ duel at 20.

He wrote down his Math findings the eve before he died – “Je n’ai pas le temps” (I have no more time) – begged his friend to send them to two foreigners (Gauss and Jacobi) for review of its importance. “Group Theory” was born in such tragic circumstances, recognized to the world only 14 years after his death.

http://en.m.wikipedia.org/wiki/%C3%89variste_Galois

Video in French and English:

Coursera: Ecole Normale Supérieure – “Introduction à La Théorie de Galois

# Free Coursera Math Class (2014)

https://www.coursera.org/course/introgalois

2014 Math Course from Coursera:   Galois Theory

Coursera is a free Online University Class taught by professors from 20+ world’s top universities, co-founded by a Singapore-born Stanford Professor Dr. Andrew Ng (ex-Rafflesian student and Computer scientist in Artificial Intelligence) with Dr. Delphne Koller.

Galois Theory is taught in the Honors / Masters Math Course in NUS (National University of Singapore, ranked the Top Asian University and 9th in the world for Mathematics in 2013)

Ecole Normale Superieure  is the ‘Teacher College’ equivalent to NTU (NIE) in Singapore. It is the top Research University in the World, producing 1/3 of the world’s Fields medalists for Math and many French Nobel Prize scientists. Évariste Galois was its student during Napoleon Revolution, after having failed the entrance exams (Concours) of Ecole Polytechniques (X) in 2 consecutive years.

However, the 19-year-old  math genius Galois was expelled too by Ecole Normale Supérieure for involving in the Revolution against the French King. Watch the latest movie “Les Misérables” by Victor Hugo, in the Parisian street barricade of student riots against the King’s soldiers, Galois was one of the student leaders.

Galois was sadly  killed at 20 in a duel by his girl friend’s fiancé – a good mathematician is not ‘necessary and sufficient condition’ to be a good shooter. The night before he died (knowing well he would be killed by a marksman), Galois wrote down the theory in 60 pages, repeatedly scribbled at the margin, “Je n’ai pas le temps” (I have no time). He asked his brother to send the papers to Jacobi and Gauss to confirm its importance. But Gauss hated Napoleon and the French,  who occupied his Germany and killed his King, refused to study the papers. Only 14 years later a  X Prof Louisville discovered the Galois Theory and Group.

Galois Theory is the most advanced theory in Algebra. The famous Fermat’s Last theorem (x^n + y^n = z^n) was proved after 350 years using Galois Theory by the Cambridge Prof Andrew Wiles in 1994.

https://www.coursera.org/course/introgalois

Another Good Math course:

https://www.coursera.org/course/functionalanalysis

French touch of graduate Math by the Engineering Ecole Centrale of Paris, the alma mater of the inventors of cars Renault, Citroën, Tower Eiffel / New York Status of Liberty.

# Ecole Polytechnique Concours 2013

French Math Exams paper is called “Composition”, it is unlike English Math paper solving different independent questions. In fact “Composition” is made up of many inter-dependent smaller questions, they together step-by-step lead to proving some Math theorems or corollary.

Before every Math composition, the French Math professor would tell the students the test scope covers all Math they learn thus far from primary school till today. Quite similar to sitting for any English language Composition, the scope of  vocaburary and grammar covers everything we learn since day 1 in primary school. Math is, after all, a “language” of science and logic.

Look at this year Ecole Polytchnique (and Ecole Normales Supérieures) Math Composition below:

http://www.ilemaths.net/maths_p-concours-polytechnique-mp-2013-01.php

[Note: My next few blogs will contain the English translation, and hopefully the solution contributed from the comments by blog readers. ]

It is notoriously famous for being very tough. It needs 2 years of preparation  after Baccalaureat (A-level) in the Classes Préparatoires (Maths Supérieures, Maths Spéciales), located not in universities but in few prestigious ancient Lycées (High Schools) selected by Napoléon eg. Lycée Louis Le Grand (Paris), Lycée Henri IV, Lycée Pierre de Fermat (Toulous), Lycée Du Park (Lyon)… taking in only the brightest Baccalaureat high-school students in Math (only 7.5% of each year High-school cohorts from Baccalaureat).

200 years ago the 19th century Math genius (father of Group Theory and Modern Algebra) Evariste Galois failed this Ecole Polytechnique “Composition” Exams twice because he was too good for the Examiners to understand him. The inventor of Topology Henri Poincaré topped in this Exams, while Charles Hermite (Galois’s 15 years junior from the same professor Richard of Lycée Louis Le Grand) was in the last position, almost failed!

Note: Ecole Normales Supérieures and Ecole Polytechnique combine their Concours entrance exams together in recent years.

# Incompetent Examiners

The two professors of the École Polytechnique (X), who demonstrated their incompetency as Examiners in the Concours (Entrance Exams),  failed the greatest math genius – Evariste Galois. Their names were:
Dinet
Lefébure de Fourcy

Distraught by the recent suicide of his father and iritated by the trivial question on Logarithms by the 2 arrogant X examiners, Galois threw the blackboard eraser at the examiner Prof. Dinet.

# Galois Theory Simplified

Galois discovered Quintic Equation has no radical (expressed with +,-,*,/, nth root) solutions, but his new Math “Group Theory” also explains:
$x^{5} - 1 = 0 \text { has radical solution}$
but
$x^{5} -x -1 = 0 \text{ has no radical solution}$

Why ?

$x^{5} - 1 = 0 \text { has 5 solutions: } \displaystyle x = e^{\frac{ik\pi}{5}}$
$\text{where k } \in \{0,1,2,3,4\}$
which can be expressed in
$x= cos \frac{k\pi}{5} + i.sin \frac{k\pi}{5}$
hence in {+,-,*,/, √ }
ie
$x_0 = e^{\frac{i.0\pi}{5}}=1$
$x_1 = e^{\frac{i\pi}{5}}$
$x_2 = e^{\frac{2i\pi}{5}}$
$x_3 = e^{\frac{3i\pi}{5}}$
$x_4 = e^{\frac{4i\pi}{5}}$
$x_5 = e^{\frac{5i\pi}{5}}=1=x_0$

=>
$\text {Permutation of solutions }{x_j} \text { forms a Cyclic Group: } \{x_0,x_1,x_2,x_3,x_4\}$

Theorem: All Cyclic Groups are Solvable
=>
$x^{5} -1 = 0 \text { has radical solutions.}$

However,
$x^{5} -x -1 = 0 \text{ has no radical solution }$
because the permutation of solutions is A5 (Alternating Group) which is Simple
ie no Normal Subgroup
=> no Symmetry!

See also: From Durian to Group

# Coset

The powerful notion of Coset was invented by Galois (l’ensemble à gauche ou à droit), but only named as Coset after 150+ yrs later by G.A. Miller in 1910.

Prove:
Coset * Coset = Coset
=> Normal Subgroup

[Hint] Proof technique: use
1) $a^{-1}$
2) e

Proof:
1) For any a ∈G, H subgroup of G,

$(Ha)(Ha^{-1})= H.(aHa^{-1})$

2) Given $H.(aHa^{-1})$ is right coset,
Choose $(aHa^{-1}) = e \in G$
$H.(aHa^{-1})= He = H$
=> $aHa^{-1} \subset H$
=> H Normal subgroup

# Galois’s 1st Public Lecture

Évariste Galois (1811–1832) (Photo credit: Wikipedia)

19-yr-old Galois’s 1st public Math Tuition class after being expelled from Ecole Normale Superieure:

“My work proceeds from a tendency to think intuitively. Dispensing the details, I prefer to leap from foresight to idea to concept…
I propose an analysis of analysis : a new, terser (abstract) algebra, where the most elevated calculations will be considered as particular case, which ultimately must be put aside for greater, more general investigations… so my foresight will be used by generations of mathematicians to advance new theories.”