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In the previous paper [15], we determined the structure of the Galois groups $\text{Gal}(K_{ur}/K)$ of the maximal unramified extensions $K_{ur}$ of imaginary quadratic number fields $K$ of conductors $\leqq 1000$ under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors $\geqq 723$) and give a table of $\text{Gal}(K_{ur}/K)$. We update the table (under GRH). For 19 exceptional fields $K$ of them, we determine $\text{Gal}(K_{ur}/K)$. In particular, for $K=𝐐\left(\sqrt{-856}\right)$, we obtain $\text{Gal}(K_{ur}/K)\cong \widetilde{S_4}\times C_5\text{and}{K}_{ur}=K_4$, the fourth Hilbert class field of $K$. This is the first example of a number field whose...

We determine the structures of the Galois groups Gal$(K_{ur}/K)$ of the maximal unramified extensions $K_{ur}$ of imaginary quadratic number fields $K$ of conductors $\leqq 420(\leqq 719$ under the Generalized Riemann Hypothesis). For all such $K$, $K_{ur}$ is $K$, the Hilbert class field of $K$, the second Hilbert class field of $K$, or the third Hilbert class field of $K$. The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class...