Class Field Theory is just like Galois Theory or the Fundamental Theorem of Algebra in the sense that the statements are useful and powerful in their own rights - you don’t really need to worry too much about their proofs before knowing how to apply them.
In this talk, I will first state the main theorems of the local and global class field theory. Then I will answer the following two questions in full detail:
1. How to find all the degree 3 abelian extensions of Q13?
2. Given a square-free integer n, how to find the minimal m such that Q(√n) ⊂ Q(ζm)?
The methods for solving the general cases will almost follow immediately.