Idelic Class Field Theory for 3-Manifolds and Number Rings

Idelic class field theory describes Abelian extensions of a number field k in terms of the idèles of k. Firstly, we construct local class field theory for the local field at each prime of k. Getting local theories together over all primes of k, we obtain idelic class field theory of k. Both local and global class field theory are derived from the arithmetic duality theorems, as we explained in Sect. 2.3 . In Sect. 7.3, we state the main results of local and global class field theory again as in Sect. 2.3 .