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

Georges Gras[1]

  • [1] Villa la Gardette Chemin Château Gagnière 38520 Le Bourg d’Oisans

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 3, page 635-654
  • ISSN: 1246-7405

Abstract

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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields K having isomorphic absolute Abelian Galois groups A K , we study any such issue for arbitrary number fields K . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p -adic obstructions coming from the global units of K . By restriction to the p -Sylow subgroups of A K and assuming the Leopoldt conjecture we show that the corresponding study is related to a generalization of the classical notion of p -rational field that we deepen, including numerical viewpoint for quadratic fields.However we obtain (Theorems 2.1 and 3.1) non-trivial information about the structure of A K , for any number field K , by application of results of our book on the p -adic global class field theory.

How to cite

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Gras, Georges. "On the structure of the Galois group of the Abelian closure of a number field." Journal de Théorie des Nombres de Bordeaux 26.3 (2014): 635-654. <http://eudml.org/doc/275767>.

@article{Gras2014,
abstract = {From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_K$, we study any such issue for arbitrary number fields $K$. We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$-adic obstructions coming from the global units of $K$. By restriction to the $p$-Sylow subgroups of $A_K$ and assuming the Leopoldt conjecture we show that the corresponding study is related to a generalization of the classical notion of $p$-rational field that we deepen, including numerical viewpoint for quadratic fields.However we obtain (Theorems 2.1 and 3.1) non-trivial information about the structure of $A_K$, for any number field $K$, by application of results of our book on the $p$-adic global class field theory.},
affiliation = {Villa la Gardette Chemin Château Gagnière 38520 Le Bourg d’Oisans},
author = {Gras, Georges},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Class field theory; Abelian closures of number fields; $p$-ramification; $p$-rational fields; Abelian profinite groups; Group extensions},
language = {eng},
month = {12},
number = {3},
pages = {635-654},
publisher = {Société Arithmétique de Bordeaux},
title = {On the structure of the Galois group of the Abelian closure of a number field},
url = {http://eudml.org/doc/275767},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Gras, Georges
TI - On the structure of the Galois group of the Abelian closure of a number field
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/12//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 3
SP - 635
EP - 654
AB - From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_K$, we study any such issue for arbitrary number fields $K$. We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$-adic obstructions coming from the global units of $K$. By restriction to the $p$-Sylow subgroups of $A_K$ and assuming the Leopoldt conjecture we show that the corresponding study is related to a generalization of the classical notion of $p$-rational field that we deepen, including numerical viewpoint for quadratic fields.However we obtain (Theorems 2.1 and 3.1) non-trivial information about the structure of $A_K$, for any number field $K$, by application of results of our book on the $p$-adic global class field theory.
LA - eng
KW - Class field theory; Abelian closures of number fields; $p$-ramification; $p$-rational fields; Abelian profinite groups; Group extensions
UR - http://eudml.org/doc/275767
ER -

References

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