This FLT (for Regular Primes) is only first part of proof up till Ideal’s Ring Theory by Kumma.
The final complete proof by Andrew Wiles (1994) used more tools: Elliptical curve + Galois Theory.
The complete proof FLT by Andrew Wiles taking him 7 years in solitude, still a short time compared to 350 years before him but failed by the grandmasters Euler/Gauss etc. Today Andrew Wiles is hailed as the greatest 21CE Mathematician, even Fields Medal gave a Special Award to him (even he was > limit age of 40 years old ).
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.
Follow up with the story #1 on FLT (Fermat’s Last Theorem), it was finally cracked 358 years later in 1994 by a British mathematician Professor Andrew Wiles in Cambridge.
The proof of FLT is itself another exciting story, a 7-year lonely task on the attic top of his Cambridge house, nobody in the world knew anything about it, until the very day when Prof Wiles gave a seemingly unrelated lecture which ended with his announcement: FLT is finally proved. The whole world was shocked!
Modular Form (MF):
Is a function which takes Complex numbers from the upper half-plane as inputs and gives Complex numbers as outputs.
MF are notable for their high level of symmetry, determined not by a single number (2π for sine) but by 2×2 Matrices of Complex numbers.
Uses:
1. Proof of the FLT
2. Investigation of Monster Group.
3. Elliptic curve = MF
4. L-function provides dictionary for translating between Analysis and Number Theory.
