#### Vol. 1, No. 1, 2013

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Imaginary quadratic fields with isomorphic abelian Galois groups

### Athanasios Angelakis and Peter Stevenhagen

Vol. 1 (2013), No. 1, 21–39
##### 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 nonisomorphic $K$ 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
absolute Galois group, class field theory, group extensions
Primary: 11R37
Secondary: 20K35
##### Milestones
Published: 14 November 2013
##### Authors
 Athanasios Angelakis Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden The Netherlands Peter Stevenhagen Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden The Netherlands