Class number one problem

In 1801, Gauss conjectured that there are exactly nine imaginary quadratic number fields with class number one. This conjecture is well known as class number one problem. In 1952, K. Heegner first solved the problem and he showed that Gauss was right about the assumption in Diophantische analysis und modulfunktionen. In this thesis, we will present a modern approach to the proof of Heegner as in D.A.Cox's book, Primes of the Form x^2 + ny^2.