페르마 마지막 정리의 증명을 이해시켜드립니다. [고등학생도 가능]

※ 본 영상에서는 Modularity Theorem, Ribet Theorem 등의 정리에 대한 직접적인 증명은 다루지 않는 대신, 그 정리들이 어떤 의미를 가지는지 이해하고, 활용하여 FLT 의 증명 흐름을 이해하는걸 목표로 하고 있습니다. 많은 검수와 검토를 했지만 틀리거나 부정확한 내용이 있을 수 있습니다.

연락/문의: [email protected]

[자료 다운로드 링크]
https://drive.google.com/file/d/1Qz-d...

[참고문헌]

※ 학부 대수학(특히 선형+현대대수)에 어느정도 익숙할 경우 Hellegouarch 의 책이 최고의 입문서라고 생각됩니다. 이 영상을 만들 때도 가장 많이 참조했습니다.

타원곡선, 모듈러 형식
Cornell, G., Silverman, J. H., & Stevens, G. (Eds.). (1997). Modular Forms and Fermat’s Last Theorem. Springer-Verlag.
Hellegouarch, Y. (2001). Invitation to the Mathematics of Fermat-Wiles. Academic Press.
Diamond, F., & Shurman, J. (2005). A First Course in Modular Forms. Springer.
Silverman, J. H. (2009). The Arithmetic of Elliptic Curves (2nd ed.). Springer.
Silverman, J. H., & Tate, J. (1992). Rational Points on Elliptic Curves. Springer.
Knapp, A. W. (1992). Elliptic Curves. Princeton University Press.
Washington, L. C. (2008). Elliptic Curves: Number Theory and Cryptography (2nd ed.). Chapman and Hall/CRC.
Koblitz, N. (1993). Introduction to Elliptic Curves and Modular Forms (2nd ed.). Springer.
Miyake, T. (1989). Modular Forms. Springer-Verlag.
Lang, S. (1995). Introduction to Modular Forms. Springer.
Neukirch, J. (1999). Algebraic Number Theory. Springer-Verlag.

논문들
Wiles, A. (1995). Modular elliptic curves and Fermat's Last Theorem. Annals of Mathematics, 141(3), 443–551.
Taylor, R., & Wiles, A. (1995). Ring-theoretic properties of certain Hecke algebras. Annals of Mathematics, 141(3), 553–572.
Ribet, K. A. (1990). On modular representations of Gal(Q̄/Q) arising from modular forms. Inventiones mathematicae, 100(2), 431–476.
Serre, J.-P. (1987). Sur les représentations modulaires de degré 2 de Gal(Q̄/Q). Duke Mathematical Journal, 54(1), 179–230.
Frey, G. (1986). Links between stable elliptic curves and certain Diophantine equations. Annales Universitatis Saraviensis. Series Mathematicae, 1(1), 1–40.

정수론
Serre, J.-P. (1973). A Course in Arithmetic. Springer-Verlag.
Burton, D. M. (2011). Elementary Number Theory (7th ed.). McGraw-Hill.
Ireland, K., & Rosen, M. (1990). A Classical Introduction to Modern Number Theory (2nd ed.). Springer.
Koblitz, N. (1994). p-adic Numbers, p-adic Analysis, and Zeta-Functions (2nd ed.). Springer.
Apostol, T. M. (1990). Modular Functions and Dirichlet Series in Number Theory (2nd ed.). Springer.

대수학
Dummit, D. S., & Foote, R. M. (2004). Abstract Algebra (3rd ed.). Wiley.
Lang, S. (2002). Algebra (Revised 3rd ed.). Springer.
Artin, M. (2011). Algebra (2nd ed.). Pearson.
Fraleigh, J. B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley.
Friedberg, S. H., Insel, A. J., & Spence, L. E. (2018). Linear Algebra (5th ed.). Pearson.

해석학
Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill.
Ahlfors, L. V. (1979). Complex Analysis (3rd ed.). McGraw-Hill.
Stein, E. M., & Shakarchi, R. (2003). Complex Analysis. Princeton University Press.
Stein, E. M., & Shakarchi, R. (2005). Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press.


0:00 인트로
2:56 FLT 증명 (n=3)
10:25 FLT 증명 (n=4)
18:47 기본 배경지식 (복소평면, 이상적분 등)
53:32 대수 배경지식 시작
54:14 Matrix (Operations, Types, Inverse, Determinant…)
1:14:31 Vector Space (Subspace, Span, Basis/Dimension…)
1:41:33 Linear Transformations (Kernal/Image, Diagonalizability, Inner Product…)
2:08:32 Group (Subgroup, Isomorphism, Homomorphism, Group Action…)
3:27:40 Ring/Field (Ideal, UFD, Ring of Polynomial)
4:09:46 앞에서 생략한 FLT (n = 3) – Lemma 증명
4:21:09 Galois Theory (Field extension, Galois group, Galois representation…)
4:53:41 p-adic analysis (p-adic valuation, Z_l, Q_l)
5:22:47 Elliptic curve basics
6:09:20 Reduction modulo p
6:20:41 Division points, Torsion subgroup
6:32:41 Weil pairing, Serre’s Theorem
6:45:14 Number of points over a finite field, Isogeny, Hasse’s Theorem, Mazur’s Theorem
7:05:10 Invariants, Minimal Equation, Semi-stability, zeta function, Artin’s theorem, Hasse-Weil L-function, Conductor
7:34:26 Modular form basics
8:36:10 The Space of Modular Forms is Finite-Dimensional, Petersson Inner Product
8:50:34 Hecke operator, Hecke Form, Basis of M_k/S_k
9:44:21 L-function of a Modular form,
9:51:36 Modularity Theorem, Ribet theorem, Tate Module, L-adic Galois representation, Ramification, Frey curve/representation
11:18:04 FLT 최종 증명


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