# A remark On Abelianized Absolute Galois Group of Imaginary Quadratic Fields

## TL;DR

This paper investigates the structure of the abelianized absolute Galois groups of imaginary quadratic fields, showing that for fixed prime class numbers, only two isomorphism types occur, with specific examples of fields sharing the same Galois group.

## Contribution

It extends known results by classifying the isomorphism types of abelianized Galois groups for imaginary quadratic fields with prime class numbers, excluding two special cases.

## Key findings

- Only two isomorphism types of Galois groups occur for fixed prime class number (excluding two special fields).
- Imaginary quadratic fields with certain discriminants have isomorphic abelianized Galois groups.
- Specific discriminants are identified where the Galois groups are isomorphic.

## Abstract

The main purpose of this paper is to extend results on isomorphism types of the abelianized absolute Galois group $\mathcal G_K^{ab}$, where $K$ denotes imaginary quadratic field. In particular, we will show that if the class number $h_K$ of an imaginary quadratic field $K$ different from $\mathbb Q(i)$, $\mathbb Q(\sqrt{-2})$ is a fixed prime number $p$ then there are only two isomorphism types of $\mathcal G_K^{ab}$ which could occur. For instance, this result implies that imaginary quadratic fields of the discriminant $D_K$ belonging to the set $\{-35, -51, -91, -115, -123, -187, -235,$ $ -267,-403, -427 \}$ all have isomorphic abelian parts of their absolute Galois groups.