The maximal unramified extensions of the imaginary quadratic number fields with class number two (Q1924210)

Let \(K=\mathbb{Q}(\sqrt d)\) be an imaginary quadratic number field with class number two, i.e. \(-d=15,20,24, \dots,403\), or 427. The author intends to determine explicitly the maximal unramified extension \(K_{ur}\) of \(K\). For this purpose, he uses essentially Odlyzko's lower bounds for discriminants [\textit{A. M. Odlyzko}, Discriminant bounds (unpublished tables), Nov. 29th, 1976], and proves that in the 3 cases of \(-d=115\), 235, 403, \(K_{ur}\) is equal to the second Hilbert class field \(K_2\) of \(K\) and in 15 other cases \(K_{ur}\) is equal to the Hilbert class field \(K_1\) of \(K\) (for only one case, \(d=-427\), do we need to assume the Generalized Riemann Hypothesis). Moreover, he obtains the result that none of the imaginary quadratic number fields with class number two has any unramified non-solvable Galois extension.