# CFT via Brauer groups vs via ideles

I am interested in the relationship between the following two versions of CFT:

Version 1: (Brauer Group Version)

Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map $inv_v:Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z}$ in a (fairly straightforward) cohomological manner. Then the short sequence: $$1\rightarrow Br(K)\rightarrow \oplus Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z}\rightarrow 1$$ where $\oplus Br(K_v)\rightarrow \mathbb{Q}/\mathbb{Z}$ is given by $\sum inv_v$, is exact.

Version 2: (Idele Version)

We can construct a map $$K^{\times}\backslash \mathbb{I}_K/\prod O_v^{\times} \to Gal(K^{ab}/K),$$ where $\mathbb{I}_K$ are the ideles associated with $K$, by $$(1,...,1,\pi_v,1,...,1)\mapsto Frob_v.$$ Furthermore, this map induces an isomorphism between the profinite completion of $K^{\times}\backslash \mathbb{I}_K/\prod O_v^{\times}$ and $Gal(K^{ab}/K)$.

My question is: How do these two formulations of Class Field Theory relate to one another. Does one imply the other and vice versa? How does one get from one statement to the other? I have never quite been able to square this circle in my mind, even though I've been exposed to CFT for years. Any help would be greatly appreciated.

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See sections 18.5--18.7 of Pierce's Associative Algebras. He proves the Brauer group exact sequence using the Artin reciprocity law in its idelic form (and the Chebotarev density theorem). –  KCd Jun 30 '12 at 4:09