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

  • SOSUKE SASAKI (a1)

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.

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

  • SOSUKE SASAKI (a1)

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