Imaginary quadratic fields with isomorphic abelian Galois groups

TL;DR

This paper computes the absolute abelian Galois groups of imaginary quadratic fields, revealing multiple non-isomorphic fields sharing identical Galois groups, thus highlighting limitations in field characterization by these groups.

Contribution

It provides a direct computation method for the Galois groups of imaginary quadratic fields and demonstrates the existence of multiple fields with identical Galois groups.

Findings

Multiple non-isomorphic imaginary quadratic fields share the same abelian Galois group.

A direct computational approach to Galois groups of imaginary quadratic fields is developed.

The results challenge the uniqueness of field characterization by their abelian Galois groups.

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.