Vol. 1, No. 1, 2013

Download this article
Download this article For screen
For printing
Recent Volumes
4: ANTS XIV
3: Hillman: Poincaré Duality
2: ANTS XIII
1: ANTS X
The Open Book Series
All Volumes
 
About the Series
Ethics Statement
Purchase Printed Copies
Author Index
 
MSP Books and Monographs
Other MSP Publications
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