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

费马大定理 Fermat’s Last Theorem (FLT): 17世纪业余数学家法国大法官费马开的一个”玩笑”, 推动350年来近代数学(Modern Mathematics)的突飞猛进。

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)

费马大法官品尚清高, 讨厌政界官僚逢场作戏的应酬, 工余爱躲在家里玩数学, 然后写信和好友(巴斯卡 Pascal, 笛卡儿 Descartes,…)讨论, 无心中发明了物理(Optics)定律, 或然率 (Probability – 和Pascal合作), 解析几何 (Analytical Geometry – 和Descartes合作)…尤其他是近代数论(Number Theory)的开山鼻祖 (他的另一个Fermat’s Little Theorem今天用在电脑密码RSA Encryption)。
他偶然读到3,000年前希腊数学家Diaophantine的书 (10世纪阿拉伯人保存, 16世纪拉丁文翻译自阿拉伯文)。他心血来潮, 在书眉写道: “我找到一个漂亮的证明这题Diaophantine Equation, 但此书旁地方太小, 不能写下”。 他死后, 儿子整理遗作而发现此书, 就成为350年来的数学疑案。


Great Math Popular Books


日本数学科普作家写的好书。深入浅出, 适合中学生读最高深数学。

結城 浩 Hiroshi Yuki (1963 -) is a Japanese Math Popular Book Writer for Secondary and High School students. In the “Galois Theory” (Chapter 10) he boldly attempted to explain to them such complicated concepts: Quotient Group, Field Extension, Group Order, Normal Sub-Group, Solvable Group …

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.

自己学习《Galois Theory》(Page 365):
— “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。”




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.


(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.


Video in French and English:

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