Stickelberger ideals of conductor p and their application

Abstract

Let p be an odd prime number and F a number field. Let K=F(ζ_p) and Δ=Gal(K/F). Let $¥mathscr{S}$_Δ be the Stickelberger ideal of the group ring Z[Δ] 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 $¥mathscr{S}$_Δ annihilates the p-ideal class group of K. In this paper, we study some properties of the ideal $¥mathscr{S}$_Δ, and check whether or not a subfield of Q(ζ_p) satisfies the above property.