If $K/\mathbb{Q}$ is an infinite algebraic extension, define as usual the class group $Cl\_K$ by the direct limit via the natural (conorm) map $Cl\_K := \lim\limits\_{\rightarrow} Cl\_F$,
where $F$ runs over all finite extensions of $\mathbb{Q}$. (I think this definition coincides with the Picard group of the ring of integers in $K$.)


For example, it is easy to see $Cl\_{\overline{\mathbb{Q}}}=0$. From Iwasawa theory if $K$ is a $\mathbb{Z}\_p$-extension of some finite extension of $\mathbb{Q}$, we know that the $p$-primary part of the class group (I will call it $p$-class group for short):
$$Cl\_K(p)=(\mathbb{Q}\_p/\mathbb{Z}\_p)^{\lambda} \times \text{ some p-group of bounded exponent}$$
If moreover $K$ is abelian over $\mathbb{Q}$, Ferrero-Washington proved that the $p$-group on the right hand side is $0$.


Now let $K=\mathbb{Q}(\mu\_{\infty})$, the field joining all the roots of unity, which is the maximal abelian extension of $\mathbb{Q}$.
**What is** $Cl\_K$ or what is $Cl\_K(p)$?


I think $Cl\_K$ has subgroups of forms, for example $H=\mathbb{Q}\_{37}/\mathbb{Z}\_{37}$, because $H$ is the $37$-class group of $F:=\mathbb{Q}(\mu\_{37^\infty})$ and one can show that the natural map from $Cl\_{F}(p)$ to $Cl\_K$ is injective, using the fact that $H$ is in the minus part (the complex conjugatation acts by $-1$) of $Cl\_F$.



What's the class group of $\mathbb{Q}^{\mathrm{ab}}$?
From Armand Brumer, [*The class group of all cyclotomic integers*](https://www.sciencedirect.com/science/article/pii/0022404981900864):



> 
> As an abelian group, $\mathrm{Pic}(O\_\infty)$ is isomorphic to a countable direct sum of copies of $\mathbb Q/\mathbb Z$.
> 
> 
> 


Here $O\_\infty$ is the ring of cyclotomic integers, and $\mathrm{Pic}$ is the Picard group - for a number ring it coincides with the class group, while for a union of an increasing tower of such it coincides with the direct limit, as in your definition.


