Number fields whose Abelian closures have isomorphic Galois groups
Résumé
Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group (arXiv:1209.6005), we study this property for arbitrary number fields. We show that such a property is probably not easily generalizable to non-imaginary quadratic fields, because of some p-adic obstructions coming from the global units. By restriction to the p-Sylow subgroups of the Galois groups, we show that the corresponding study is related to the classical notion of p-rational fields. All this is an application of results published in our Springer book on class field theory.
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