Some examples of quadratic fields with finite non-solvable maximal unramified extensions II

Abstract

Let K be a number field and $$K_\mathrm{ur}$$ be the maximal extension of K that is unramified at all places. In a previous article (Kim, J Number Theory 166:235–249, 2016), the first author found three real quadratic fields K such that $$\mathrm {Gal}(K_\mathrm{ur}/K)$$ is finite and non-abelian simple under the assumption of the generalized Riemann hypothesis (GRH). In this article, we extend the methods of Kim (2016) and identify more quadratic number fields K such that $$\mathrm {Gal}(K_\mathrm{ur}/K)$$ is a finite nonsolvable group and also explicitly calculate their Galois groups under the assumption of the GRH. In particular, we find the first imaginary quadratic field with this property.