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

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  09 July 2018

SOSUKE SASAKI
SOSUKE SASAKI*
Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo, Japan email [email protected]
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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.

MSC classification

Primary: 11R29: Class numbers, class groups, discriminants 11R32: Galois theory 11R45: Density theorems
Type
Article
Information
Nagoya Mathematical Journal , Volume 237 , March 2020 , pp. 166 - 187
Copyright
© 2018 Foundation Nagoya Mathematical Journal  

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