Stickelberger ideals of conductor p and their application

Let p be an odd prime number and F a number field. Let K=F(\zeta_p) and \Delta=\mathrm{Gal}(K/F) . Let \mathscr{S}_{\Delta} be the Stickelberger ideal of the group ring \mathbf{Z}[\Delta] defined in the previous paper [8]. As a consequence of a p -integer version of a theorem of McCulloh [15], [16], it follows that F has the Hilbert-Speiser type property for the rings of p -integers of elementary abelian extensions over F of exponent p if and only if the ideal \mathscr{S}_{\Delta} annihilates the p -ideal class group of K . In this paper, we study some properties of the ideal \mathscr{S}_{\Delta} ,and check whether or not a subfield of \mathbf{Q}(\zeta_p) satisfies the above property.