On the splitting of the Hilbert class field
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.
scientific article; zbMATH DE number 98397
scientific article