Abstract
Let K be a finite Galois extension of a number field F and K′ the Hilbert class field of K. We consider the splitting of the group extension defined by the Galois groups of the tower of fields F ⊂ K ⊂ K′. We give necessary and sufficient conditions for the extension to split when K is an unramified abelian extension of an imaginary quadratic field. We then consider the splitting of the extension for some imaginary biquadratic fields K.
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