Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

Imaginary quadratic fields with isomorphic abelian Galois groups

Athanasios Angelakis and Peter Stevenhagen

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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