1. Imaginary quadratic fields with isomorphic abelian Galois groups
- Author
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Athanasios Angelakis, Peter Stevenhagen, Lithe and fast algorithmic number theory (LFANT), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Mathematical institute, Universiteit Leiden [Leiden], Everett W. Howe and Kiran S. Kedlaya, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Universiteit Leiden
- Subjects
Pure mathematics ,Profinite group ,Mathematics - Number Theory ,group extensions ,Mathematics::Number Theory ,010102 general mathematics ,Galois group ,010103 numerical & computational mathematics ,Absolute Galois group ,Basis (universal algebra) ,Algebraic number field ,16. Peace & justice ,01 natural sciences ,11R37 (Primary) 20K35 (Secondary) ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,2000 Mathematics Subject Classification. Primary 11R37 ,Secondary 20K35 ,class field theory ,Character (mathematics) ,absolute Galois group ,Class field theory ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Abelian group ,Mathematics - 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., Comment: 17 pages; to appear in the proceedings volume of ANTS-X, San Diego 2012
- Published
- 2012