On the splitting of the Hilbert class field

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.