# Theorem of class field theory in appendix of the book 'Cyclotomic fields' by Washington

I came across the following theorem of class field theory in the appendix of the book 'Cyclotomic fields' by Washington. I could not find it in Milne, Cassels Frohlich, Neukirch or Artin Tate.

Theorem 14. (a) If $$K/k$$ is abelian, then there is a closed subgroup $$H$$ with $$D_k\subseteq H\subseteq C_k$$, such that $$C_k/H\simeq\mathrm{Gal}(K/k).$$ The prime $$\mathfrak{p}$$ is unramified $$\Longleftrightarrow k^\times U_\mathfrak{p}/k^\times\subseteq H$$.

(b) Given a closed subgroup $$H$$ with $$D_k\subseteq H\subseteq C_k$$ (equivalently, $$C_k/H$$ is totally disconnected), there is a unique abelian extension corresponding to $$H$$, as in (a).

Where can I find proof and discussion on this theorem?

## 1 Answer

this is called the existence theorem and I think it is mentioned in both Milne book and Cassels Frohlich, possibly with a different language. but It is discussed in length in the Bonn lecture of Neukirch https://www.springer.com/gp/book/9783642354366. the part about the ramification comes from local class field theory and compatibility of local and global class field theory.

• The complete argument seems to be in Artin Tate. But the argument seems rather long, is there a short and crisp exposition? Feb 6, 2021 at 16:14
• the proof is a little technical, but the main idea is that: 1) if $H'\in H$ and there is an associated abelian extension for $H'$ then there is an abelian extension associated to $H$ this is an easy consequence of reciprocity theorem2)then you have to compute the norm subgroup of kummer extensions and show that every closed subgroup contain norm group of a Kummer subgroup. or course in function fields you have to do a little more work because kummer extensions are not enough
– ali
Feb 6, 2021 at 16:47
• if you are familiar with basic properties of ideles then the proof in bonn lectures is less than two pages, and I think every step is basically the only sensible thing you can do if you want to follow strategy of my last comment
– ali
Feb 6, 2021 at 16:52