Nagoya Mathematical Journal ( IF 0.638 ) Pub Date : 2018-07-09 , DOI: 10.1017/nmj.2018.16
SOSUKE SASAKI

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.

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