Let $k$ be an imaginary quadratic field with $\operatorname{Cl}_{2}(k)\simeq V_{4}$. It is known that the length of the Hilbert $2$-class field tower is at least $2$. Gerth (On 2-class field towers for quadratic number fields with$2$-class group of type$(2,2)$, Glasgow Math. J. 40(1) (1998), 63–69) calculated the density of $k$ where the length of the tower is $1$; that is, the maximal unramified $2$-extension is a $V_{4}$-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.

中文翻译：

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最大非分幅 2-延展的 GALOIS 群在虚二次域上的分布

让$k$是一个虚构的二次场$\operatorname{Cl}_{2}(k)\simeq V_{4}$. 已知希尔伯特长度$2$- 级场塔至少$2$. 格特 (在二次数场的 2 类场塔上$2$-类组类型$(2,2)$，格拉斯哥数学。J。40(1) (1998), 63–69) 计算了$k$塔的长度在哪里$1$; 也就是说，最大的未分枝$2$-扩展是一个$V_{4}$-延期。在本文中，我们将把这个结果扩展到小度数的广义四元数、二面体和半二面体扩展。