A note on the splitting of the Hilbert class field

Abstract

Let K be a finite Galois extension of a number field k and H the Hilbert class field of K. We study whether the group extension defined by the Galois groups of the tower of fields H ⊇ K ⊇ k splits. We first rederive earlier theorems of Wyman and Gold by a new method and show how this method can slightly extend their results. We then give a necessary condition for splitting and show that in many cases this extension cannot split.