Foreword
1 Paths
1.1 Diophantus and his Arithmetica
1.2 Translations of Diophantus
1.3 Fermat
1.4 Infinite Descent
1.5 Fermat’s “Theorem” in Degree 4
1.6 The Theorem of Two Squares
1.6.1 A Modern Proof
1.6.2 “Fermat-Style” Proof of the Crucial Theorem
1.6.3 Representations as Sums of Two Squares
1.7 Euler-Style Proof of Fermat’s Last Theorem for n=3
1.8 Kummer, 1847
1.8.1 The Ring of Integers of Q(ξ)
1.8.2 A Lemma of Kummer on the Units of Z[ξ]
1.8.3 The Ideals of Z[ξ]
1.8.4 Kummer’s Proof (1847)
1.8.5 Regular Primes
1.9 The Current Approach
Exercises and Problems
2 Elliptic Functions
2.1 Elliptic Integrals
2.2 The Discovery of Elliptic Functions in 1718
2.3 Euler’s Contribution (1753)
2.4 Elliptic Functions: Structure Theorems
2.5 Weierstrass-Style Elliptic Functions
2.6 Eisenstein Series
2.7 The Weierstrass Cubic
2.8 Abel’s Theorem
2.9 Loxodromic Functions
2.10 The Function ρ
2.11 Computation of the Discriminant
2.12 Relation to Elliptic Functions
Exercises and Problems
3 Numbers and groups
3.1 Absolute Values on Q
3.2 Completion of a Fequipped with an Absolute Value
3.3 The Field of p-adic Numbers
3.4 Algebraic Closure of a Field
3.5 Generalities on the Linear Representations of Groups
3.6 Galois Extensions
3.6.1 The Galois Correspondence
3.6.2 Questions of Dimension
3.6.3 Stability
3.6.4 Conclusions
3.7 Resolution of Algebraic Equations
3.7.1 Some General Principles
3.7.2 Resolution of the Equation of Degree Three
Exercises and Problems
4 Elliptic Curves
4.1 Cubics and Elliptic Curves
4.2 B´ezout’s Theorem
4.3 Nine-Point Theorem
4.4 Group Laws on an