  Following a paper by Athanasios Angelakis and Peter Stevenhagen on the
determination of imaginary quadratic fields having the same absolute Abelian
Galois group A, we study this property for arbitrary number fields. We show
that such a property is probably not easily generalizable, apart from imaginary
quadratic fields, because of some p-adic obstructions coming from the global
units. By restriction to the p-Sylow subgroups of A, we show that the
corresponding study is related to a generalization of the classical notion of
p-rational fields. However, we obtain some non-trivial information about the
structure of the profinite group A, for every number field, by application of
results published in our book on class field theory. This version corrects a
technical error discovered by Peter Stevenhagen in a lemma of their first draft
(arXiv:1209.6005) as well as in the previous versions of our paper reproducing
this lemma; it will also be corrected in their final paper to appear in the
proceedings volume of ANTS-X, San Diego 2012. This does not modify the nature
of the results.
