Let $K$ be the imaginary quadratic field obtained by joining $\sqrt{-1}$ to the field of rational numbers $Q$. I would like to describe the extension $K^{ab}/Q^{ab}$, where for $F$ a number field, $F^{ab}$ denotes its maximal abelian extension (everything is taking place inside a big fixed field...).


More precisely I would like to know the Galois group and the ramification properties of such extension. Is this possible/easy? I suppose one should look at the kernel of the norm map between Idele class groups $N\_{K/Q}:I\_K\rightarrow I\_Q$. But at the moment it is not clear to me how to get the answer. Any hint or comment would be appreciated. Thanks.


EDIT: Probably the idele class group of a number field $F$ should be denoted by $J\_F$. Or by anything other than $I\_F$...



An explicit computation in class field theory
Given that you want to know the structure of the Galois group and ramification, I think that you are best off working with the kernel of the norm map between connected components of idele class groups, as you yourself suggest.


These groups are very explicit: for $K := \mathbb Q(i)$, one obtains $\hat{\mathcal O}\_K^{\times}/\{\pm 1,\pm i\}$,
and for $\mathbb Q$ one obtains $\hat{\mathbb Z}.$ (Here $\hat{}$ denotes the profinite
completion.) Apart from the diagonally embedded $\{\pm 1,\pm i\}$ quotient in the
group for $K$, both groups factor as a product over primes, and the norm map is given
component wise. 


So the kernel of the norm map is equal to
$$\bigl(\prod\_p (\mathcal O\_K\otimes\_{\mathbb Z}\mathbb Z\_p)^{\times, \text{Norm } = 1}\bigr)/
\{\pm 1,\pm i\}.$$


This should be explicit enough to answer any particular question you have.


