Three Ancient Greek Problems : a) Trisect an angle“, b) Doubling the Cube, c) Square a Circle

Field Theory

Abel

Galois

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.

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 “Insolvabilityof 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.

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.

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).

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.