I asked this question about a week ago here http://math.stackexchange.com/questions/288955/splitting-the-exact-sequence-of-the-idele-class-group, but got no answer so I thought I'd aske here and see if I can get some help.

I was wondering if there are any conditions on a number field $K$ such that $$0 \longrightarrow K^{\times} \longrightarrow \mathbb{A}_{K}^{\times} \longrightarrow C_K \longrightarrow 0, $$ is a splits as a sequence of multiplicative abelian groups (here I have done the usual diagonal embedding of $K^{\times}$ into $\mathbb{A}_{K}^{\times}$)

I thought about looking at $H^{2}(C_K,K^{\times})$, but I'm not quite sure how to compute it.

Thank you.

`$\prod_{v|\infty}K_v^{\times} \times \prod_{v\nmid\infty} O_{K,v}^{\times}$`

, so $r'(O_K^{\times})=1$. But $r(O_K^{\times})=O_K^{\times}$, so $K$ is imag. quadratic. – user30379 Feb 2 '13 at 17:02