On the Galois module structure of cyclic Kummer extensions

Let p be an odd prime number. Let K/k be a cyclic totally ramified Kummer extension of degree p(n) with the Galois group G and assume K has a Kummer generator satisfying some conditions. Let D and a be the rings of integers in K and k, respectively. In case k is a p-adic number field, let D-1 be the ring of integers in the subextension of K/k of degree p(l) over k. We obtain conditions that all subrings D-1 of D = D-n (0 < 1 <= n) are free over associated orders U-1 of D-1, and we prove that U = U-n is stable under the action of automorphisms of G. In case k is an algebraic number field, for tamely ramified abelian extensions K/k, McCulloh gave the characterization of the set R(oG) of realizable Galois module classes cl(D) and proved that R(o vertical bar mu(E)vertical bar) contains Cl(o vertical bar mu(E)vertical bar)(Sc), where o vertical bar mu(E)vertical bar is a certain quotient ring of oG and Sc is the Stickelberger ideal. For cyclic wildly ramified Kummer extensions K/k, we obtain the relation between two sets R(o vertical bar mu(E)vertical bar) and Cl(o vertical bar mu(E)vertical bar)(Sc) which is the analogous result to his result. (C) 2008 Elsevier Inc. All rights reserved.