A brief summary of the main statements of class field theory

0. Profinite completions of topological groups
1. Local class field theory
1.1. Notation associated to a discrete valuation ring
1.2. Local fields
1.3. The local Artin homomorphism
1.4. Functoriality
1.5. Finite abelian extensions
2. Global class field theory (via ideles)
2.1. Global fields
2.2. The adele ring
2.3. The idele group and idele class group
2.4. The global Artin homomorphism
2.5. Functoriality
2.6. Finite abelian extensions
2.7. Connection between the global and local Artin homomorphisms
2.8. Moduli
2.9. Ray class groups and ray class fields
3. Global class field theory (via ideals)
3.1. Classical ray class groups
3.2. The classical Artin homomorphism
3.3. The main theorems
3.4. Comparison of ideal groups and idele groups
4. An introduction to an introduction to the Langlands program
5. Further reading
References