On the splitting of the Hilbert class field
type: Item
instance of: scholarly article
title: On the splitting of the Hilbert class field
zbMATH Open document ID: 0778.11063
DOI: 10.1016/0022-314X(92)90099-B
author: Robert J. Bond
published in: Journal of Number Theory
publication date: 1993-01-17T00:00:00Z
review text: Let \(K/F\) be a finite Galois extension of number fields and \(K'\) denote the Hilbert class field of \(K\). \textit{G. Cornell} and \textit{M. Rosen} [Queen's Pap. Pure Appl. Math. 54, 287-308 (1980; Zbl 0471.12012)] have given necessary conditions for the sequence \[ 1 \longrightarrow G(K'/K) \longrightarrow G(K'/F) \longrightarrow G(K/F) \longrightarrow 1 \] to split. The author gives another proof of their result and shows the condition is not sufficient. Moreover, necessary and sufficient conditions are given for the sequence to split when \(F\) is an imaginary quadratic field and \(K\) is a subfield of the Hilbert class field \(F'\) of \(F\). In particular, if \([K:F]\) is odd the conditions are that \(G(K/F)\) be cyclic and no ideal class of \(G(F'/K)\) becomes trivial in \(K\). A slightly more complicated condition is given when \([K:F]\) is a power of 2. The latter result is applied to several examples where \(K\) is a bicyclic biquadratic field. For example, if \(p\equiv q\equiv 1\pmod 4\) are primes, \(F=\mathbb{Q}(\sqrt{-pq})\) and \(K=\mathbb{Q}(\sqrt{p},\sqrt{-q})\) then the sequence splits if and only if \(p\) and \(q\) are quadratic nonresidues of each other.
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 98397
zbMATH Keywords: splitting
zbMATH Keywords: Galois groups
zbMATH Keywords: group extensions
zbMATH Keywords: Hilbert class field
zbMATH Keywords: imaginary quadratic field
zbMATH Keywords: bicyclic biquadratic field
reviewed by: Charles J. Parry
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 98397