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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 $I_F$...

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3 Answers 3

up vote 16 down vote accepted

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.

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1  
That's really nice! I feel slightly embarrassed to admit it, but I had never thought about it this way: your local factors then give a very explicit description of the subgroup of image of Galois in $\text{GL}_2(\mathbb{Z}_p)$ on the Tate modules of the elliptic curve that fixes the cyclotomic extension. I somehow always assumed that this subgroup of $\text{GL}_2(\hat{\mathbb{Z}})$ would be much messier to write down. –  Alex B. Oct 20 '10 at 2:40
    
@Emerton: Thanks for your answer! –  unknown Oct 21 '10 at 0:59
    
@Alex: I do not understand your comment. What does the Tate module of some elliptic curve have to do with the group written by Emerton above? –  unknown Oct 21 '10 at 1:10

In your particular case, $K^{ab}$ is completely understood, but your field is one of the very few for which such an explicit class field theory is known, so you got lucky.

I don't know your background, but to understand the answer you need to know something about the theory of complex multiplication. What I am going to say works for any imaginary quadratic field. The field $K^{ab}$ is generated by so-called Weber functions, usually just given by the $x$-coordinates of torsion points on any elliptic curve that has complex multiplication by the ring of integers of $K$. Actually, in your particular case, you are looking e.g. at the elliptic curve $y^2 = x^3+x$ and the maximal abelian extension is just generated by the torsion points (always true when $K$ has class number one).

You can read up on this in Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", Chapter II. Have a look particularly at example 5.8.1.

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Use the theory of complex multiplication. $K^{ab}$ is the field generated by the torsion points of $y^2=x^3+x$.

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Thanks! Can I then answer my question with this information? –  unknown Oct 20 '10 at 0:57

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