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 AK. The first examples of nonisomorphic K having isomorphic AK 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 AK. In this paper, we provide a direct “computation” of the profinite group AK 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
Mathematical Subject Classification 2010
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