My previous question about the theorem (apparently due to Dedekind -- thanks, Arturo Magidin!) that the ring $\overline{\mathbb{Z}}$ of all algebraic integers is a Bezout domain got me thinking about rings of integers in infinite algebraic extensions of $\mathbb{Q}$.

As I mentioned here, I was thinking about Kaplansky's formulation/generalization of Dedekind's theorem: if $R$ is a Dedekind domain with fraction field $K$ such that for all finite degree field extensions $L/K$ the ideal class group of the integral closure $R_L$ of $R$ in $L$ is a torsion abelian group, then the integral closure $S$ of $R$ in $\overline{K}$ is a Bezout domain.

Since we know (by nefarious arithmetic arguments...) that the Picard group of the ring of integers of a number field is finite, Kaplansky's theorem applies with $R = \mathbb{Z}$ and gives Dedekind's theorem.

Suppose we now look at Kaplansky's proof and see if it gives more: do we need to go all the way up to $\overline{\mathbb{Q}}$ in order to show that the ring of integers is a Bezout domain?

Indeed not. The key property of the ring $S$ used in the proof is that it is **root closed**, i.e., closed under extraction of $n$th roots, for all $n \in \mathbb{Z}^+$. There are lots of smaller -- but still infinite -- algebraic extensions of $\mathbb{Q}$ which have this property: the smallest is $\mathbb{Q}^{\operatorname{solv}}$, the maximal solvable extension of $\mathbb{Q}$.

But what about the ring of integers of $\mathbb{Q}^{\operatorname{ab}}$, the maximal abelian extension of $\mathbb{Q}$? Is this a Bezout domain? If not, what is its Picard group?

In general, if $R$ is a Dedekind domain with fraction field $K$, $L/K$ is any algebraic field extension and $S$ is the integral closure of $R$ in $L$, then if $[L:K]$ is infinite $S$ need not be a Dedekind domain but is always a Prufer domain: every finitely generated ideal is invertible. In particular $S$ is a Bezout domain iff $\operatorname{Pic}(S) = 0$. So one may ask:

What is known about the Picard groups of rings of integers of infinite algebraic extensions of $\mathbb{Q}$? What, if anything, is the connection to more classical algebraic number theory?