DISTRIBUTION OF GALOIS GROUPS OF MAXIMAL UNRAMIFIED 2-EXTENSIONS OVER IMAGINARY QUADRATIC FIELDS

Abstract

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.