# Imaginary quadratic fields with isomorphic abelian Galois groups

1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : 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.
Keywords :
Type de document :
Communication dans un congrès
Everett W. Howe and Kiran S. Kedlaya. ANTS X - Tenth Algorithmic Number Theory Symposium, Jul 2012, San Diego, United States. Mathematical Sciences Publisher, 1, pp.21-39, 2013, ANTS X - Proceedings of the Tenth Algorithmic Number Theory Symposium. 〈10.2140/obs.2013.1.21〉
Liste complète des métadonnées

https://hal.inria.fr/hal-00751883
Contributeur : Angelakis Athanasios <>
Soumis le : mercredi 14 novembre 2012 - 14:24:14
Dernière modification le : jeudi 11 janvier 2018 - 06:22:36

### Citation

Athanasios Angelakis, Peter Stevenhagen. Imaginary quadratic fields with isomorphic abelian Galois groups. Everett W. Howe and Kiran S. Kedlaya. ANTS X - Tenth Algorithmic Number Theory Symposium, Jul 2012, San Diego, United States. Mathematical Sciences Publisher, 1, pp.21-39, 2013, ANTS X - Proceedings of the Tenth Algorithmic Number Theory Symposium. 〈10.2140/obs.2013.1.21〉. 〈hal-00751883〉

### Métriques

Consultations de la notice