I am learning Class Field Theory by reading Milne's notes and Neukirch's book. There is a proof I can't find.
Let $K$ be a number field. One constructs the map $I_K \rightarrow G_K^\text{ab}$ using local class field theory then shows that it factors through $C_K/C_K^0 = I_K/ \overline{(K^\times.K_\infty^\times)^\circ} \to G_K^\text{ab}$. My question is:
How to prove that this map is injective, surjective and that it is an homeomorphism?
In Milne's note (remark 5.7, p. 174), he claims that is is bijective without proving it (or am I missing something?).