Chapter 1 Fermat 1.1 Fermat and his'Last Theorem' 1.2 Pythagorean triangles 1.3 How to find Pythagorean triples 1.4 The method of infinite descent 1.5 The case n=4 of the Last Theorem 1.6 Fermat's one proof 1.7 Suns of two squares and related topics 1.8 Perfect numbers and Fermat's theorem 1.9 Pell's equation 1.10 Other number-theoretic discoveries of Fermat Chapter 2 Euler 2.1 Euler and the case n=3 2.2 Euler's proof of the case n=3 2.3 Arithmetic of surds 2.4 Euler on sums of two squares 2.5 Remainder of the proof when n=3 2.6 Addendum on sums of two squares Chapter 3 From Euler to Kummer 3.1 Introduction 3.2 Sophie Germain's theorem 3.3 The case n=5 3.4 The cases n=14 and n=7 Chapter 4 Kummer's theory of ideal factors 4.1 The events of 1847 4.2 Cyclotomic integers 4.3 Factorization of primes p*1 mod λ 4.4 Computations when p*1 mod λ 4.5 Periods 4.6 Factorization of primes p≠1 mod λ 4.7 Computations when p≠1 mod λ 4.8 Extension of divisibility test 4.9 Prime divisors 4.10 Multiplicities and the exceptional prime 4.11 The fundamental theorem 4.12 Divisors 4.13 Terminology 4.14 Conjugations and the norm of a divisor 4.15 Summary Chapter 5 fermat's Last Theorem for regular primes 5.1 Kummer's remarks on quadratic integers 5.2 Equivalence of divisors in a special case 5.3 The class number 5.4 Kummer's two conditions 5.5 The proof for regular primes 5.6 Quadratic reciprocity Chapter 6 Determination of the class number 6.1 Introduction 6.2 The Euler product formula 6.3 First steps 6.4 Reformulation of the right side 6.5 Dirichlet's evaluation of L(l,x) 6.6 The limit of the right side 6.7 The nonvanishing of L-series 6.8 Reformulation of the left side 6.9 Units:The first few cases 6.10 Units:the general case 6.11 Evaluation of the integral 6.12 Comparison of the integral and the sum 6.13 The sum over other divisor classes 6.14 The class number formula 6.15 proof that 37 is irregular 6.16 Divisibility of the first factor by λ 6.17 Divisibility of the second factor by λ 6.18 Kummer's lamma 6.19 Summary Chapter 7 Divisor theory for quadratic integers 7.1 The prime divisors 7.2 The divisor theory 7.3 The sign of the norm 7.4 Quadratic integers with given divisors 7.5 Validity of the cyclic method 7.6 The divisor class group:examples 7.7 The divisor class group:a general theorem 7.8 Euler's theorems 7.9 Genera 7.10 Ambiguous divisors 7.11 Gauss's second proof of quadratic reciprocity Chapter 8 Gauss's theory of binary quadratic forms 8.1 Other divisor class groups 8.2 Alternative view of the cyclic method 8.3 The correspondence between divsors and binary quadratic forms 8.4 The classification of forms 8.5 Examples 8.6 Gauss's composition of forms 8.7 Equations of degree 2 in 2 variables Chpater 9 Dirichlet's class number formula 9.1 The euler product formula 9.2 First case 9.3 Another case 9.4 D≡1 mod 4 9.5 Evaluation of Σ(^D_n)1/n 9.6 Suborders 9.7 Primes in arithmetic progressions Appendix:The natural numbers A.1 Basic properties A.2 primitive roots mod p Answers to exercises Bibliography Index