On the structure of the Galois group of the Abelian closure of a number field

TL;DR

This paper investigates the structure of the Galois group of the Abelian closure of number fields, highlighting obstructions and structural properties, especially for imaginary quadratic fields, using class field theory results.

Contribution

It extends the study of Galois groups of Abelian closures to arbitrary number fields and relates the problem to p-rational fields and p-Sylow subgroups.

Findings

Obstructions prevent generalization beyond imaginary quadratic fields.

Structural insights into the Galois group using class field theory.

Relation to p-rational fields and p-Sylow subgroups.

Abstract

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.