  In 1976, Onabe discovered that, in contrast to the Neukirch-Uchida results
that were proved around the same time, a number field $K$ is not completely
characterized by its absolute abelian Galois group $A_K$. The first examples of
non-isomorphic $K$ having isomorphic $A_K$ were obtained on the basis of a
classification by Kubota of idele class character groups in terms of their
infinite families of Ulm invariants, and did not yield a description of $A_K$.
In this paper, we provide a direct `computation' of the profinite group $A_K$
for imaginary quadratic $K$, and use it to obtain many different $K$ that all
have the same minimal absolute abelian Galois group.
