Ⅰ.Quasi-historical introduction
The cases n=2 and n=4.The Parisian Academy in the 1840s.
Notes:Some details.Descent.Algebraic numbers and integers.
Ⅱ.Remarks on unique factorization
A digression.
Notes:Continued fractions.Plagiarism.
Ⅲ.Elementary methods
Sophie Germain,Abel's formulas,Mirimanoff-Wieferich,...
Notes:Fermat's Theorem.Bernoulli numbers.Euler-Maclaurin.
Pseudoprimes.Fermat numbers.Mersenne primes.Cranks.
Ⅳ.Kummer's arguments
Proof of the FLT for regular primes.
Notes:Some remarks for undergraduates on elementary algebra.
Equivalence relations.
Ⅴ.Why do we believe Wiles? More quasi-history
Rantings.Work on the FLT this century.
Notes:Euler's conjecture.The growing of the "gap".
Ⅵ.Diophantus and Fermat
What the study of diophantine equations is really all about.
Notes:The chord and tangent method.Examples.
Ⅶ.A child's introduction to elliptic functions
For a precocious child.
Notes:Discriminants.
Ⅷ.Local and global
Some remarks on p-adic numbers.
Notes:The Riemann ζ-function.Much more on p-adic numbers.
Ⅸ.Curves
Particularly,about elliptic curves.
Notes:Minimal model.Semisimplicity of the Frey curve.
Birational equivalence.
Ⅹ.Modular forms
Some formulas and assertions.
Notes:More formulas.The discriminant function.
Ⅺ.The Modularity Conjecture
An attempt at an explanation.
Notes:What's in a name?
Ⅻ.The functional equation
Poisson summation;θ-functions.
Notes:Details.Hecke operators.
ⅩⅢ.Zeta functions and L-series
Introduction to the Birch-Swinnerton-Dyer Conjectures.
Notes:Hasse's Theorem.
ⅩⅣ.The ABC-Conjecture
Darmon and Granville's Generalized Fermat Equation.
Notes:Hawkins primes.The Generalized Fermat Conjecture.
ⅩⅤ.Heights
Remarks on the Mordell-Weil Theorem.
Notes:Lehmer's Question.Elliptic curves of high rank.
ⅩⅥ.Class number of imaginary quadratic number fields
The proof of Goldfeld-Gross-Zagier.
Notes:Composition of quadratic forms.Tate-Shafarevitch group.
Jacobian.Heegner points.
ⅩⅦ.Wiles'proof
Not the commutative algebra,of course.
Notes:Some details.
Appendices
A.Remarks on Fermat's Last Theorem
For those who only want to pretend to have read the rest of this book.
B."The Devil and Simon Flagg",by Arthur Porges
The devil fails where Wiles will succeed.
C."Math Riots Prove Fun Incalculable",by Eric Zorn
Is the FLT truly as important as sport?
Index
编辑手记
