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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?

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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.

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  • $\begingroup$ The complete argument seems to be in Artin Tate. But the argument seems rather long, is there a short and crisp exposition? $\endgroup$
    – mathemather
    Feb 6, 2021 at 16:14
  • $\begingroup$ 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 $\endgroup$
    – ali
    Feb 6, 2021 at 16:47
  • $\begingroup$ 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 $\endgroup$
    – ali
    Feb 6, 2021 at 16:52

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