Some examples of quadratic fields with finite nonsolvable maximal unramified extensions II

TL;DR

This paper identifies additional quadratic number fields with finite, nonabelian simple Galois groups of their maximal unramified extensions, explicitly calculating these groups under the assumption of the Generalized Riemann Hypothesis.

Contribution

It extends previous work by finding more quadratic fields with finite nonsolvable Galois groups of their unramified extensions and explicitly determining these groups assuming GRH.

Findings

Identified new quadratic fields with finite nonsolvable Galois groups

Explicit calculations of Galois groups under GRH

Extended classification of quadratic fields with specific unramified extension properties

Abstract

Let $K$ be a number field and $K_{ur}$ be the maximal extension of $K$ that is unramified at all places. In a previous article, the first author found three real quadratic fields $K$ such that $Gal(K_{ur}/K)$ is finite and nonabelian simple under the assumption of the GRH(Generalized Riemann Hypothesis). In this article, we will identify more quadratic number fields $K$ such that $Gal(K_{ur}/K)$ is a finite nonsolvable group and also explicitly calculate their Galois groups under the assumption of the Generalized Riemann Hypothesis.