Imaginary quadratic fields with isomorphic abelian Galois groups

الأصل · EN

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 non-isomorphic 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.