Fermat's last theorem : A genetic introduction to algebraic number theory

Includes index

Includes Bibliography (p. 403-407) and references

Fermat : Fermat and his "last theorem" ; Pythagorean triangles ; How to find Pythagorean triples ; The method of infinite descent ; The case n=4 of the last theorem ; Fermat's one proof ; Sums of two squares and related topics ; Perfect numbers and Fermat's theorem ; Pell's equation ; Other number-theoretic discoveries of Fermat
Euler : Euler and the case n=3 ; Euler's proof of the case n=3 ; Arithmetic of surds ; Euler on sums of two squares ; Remainder of the proof when n=3 ; Addendum on sums of two squares
From Euler to Kummer : Sophie Germain's theorem ; The case n=5 ; The cases n=14 and n=7
Kummer's theory of ideal factors : The events of 1847 ; Cyclotomic integers ; Factorization of primes p[] 1 mod [] ; Periods ; Factorization of primes p [] 1 mod [] ; Computations when p [] 1 mod [] ; Extension of the divisibility test ; Prime divisors ; Multiplicities and the exceptional prime ; The fundamental theorem ; Divisors ; Terminology ; Conjugations and the norm of a divisor
Fermat's last theorem for regular primes : Kummer's remarks on quadratic integers ; Equivalence of divisors in a special case ; The class number ; Kummer's two conditions ; The proof for regular primes ; Quadratic reciprocity
Determination of the class number : The Euler product formula ; First steps ; Reformation of the right side ; Dirichlet's evaluation of L(1, x) ; The limit of the right side ; The nonvanishing of L-series ; Reformation of the left side ; Units: the first few cases ; Units: the general case ; Evaluation of the integral ; Comparison of the integral and the sum ; The sum over other divisor classes ; The class member formula ; Proof that 37 is irregular ; Divisibility of the first factor by [] ; Divisibility of the second factor by [] ; Kummer's lemma
Divisor theory for quadratic integers : The prime divisors ; The divisor theory ; The sign of the norm ; Quadratic integers with given divisors ; Validity of the cyclic method ; The divisor class group: examples ; The divisor class group: a general theorem ; Euler's theorems ; Genera ; Ambiguous classes ; Gauss's second proof of quadratic reciprocity
Gauss's theory of binary quadratic forms : Other divisor class groups ; Alternative view of the cyclic method ; The correspondence between divisors and binary quadratic forms ; The classification of forms ; Examples ; Gauss's composition of forms ; Equation of degree 2 in 2 variables
Dirichlet's class number formula : The Euler product formula ; First case ; Another case ; D [] 1 mod 4 ; Evaluation of []([])[] ; Suborders ; Primes in arithmetic progression

This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The book also covers in detail the application of Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.