Class Field Theory and Elliptic Curves with Complex Multiplication

Class field theory is a branch of algebraic number theory which has the purpose of studying and classifying abelian extensions of fields. The work starts with a detailed study of this theory based on a cohomological approach which leads to the statements and the proofs of the main theorems both in the local and global cases. Then, after a brief introduction on elliptic curves with complex multiplication, the main goal of the thesis is to study their relation with class field theory in the particular case of quadratic imaginary fields.