# 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.
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Conference papers

https://hal.inria.fr/hal-00751883
Contributor : Angelakis Athanasios <>
Submitted on : Wednesday, November 14, 2012 - 2:24:14 PM
Last modification on : Thursday, January 11, 2018 - 6:22:36 AM

### Citation

Athanasios Angelakis, Peter Stevenhagen. Imaginary quadratic fields with isomorphic abelian Galois groups. ANTS X - Tenth Algorithmic Number Theory Symposium, Jul 2012, San Diego, United States. pp.21-39, ⟨10.2140/obs.2013.1.21⟩. ⟨hal-00751883⟩

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