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 pn with the Galois group G and assume K has a Kummer generator satisfying some conditions. Let and be the rings of integers in K and k, respectively. In case k is a -adic number field, let be the ring of integers in the subextension of K/k of degree pl over k. We obtain conditions that all subrings of (0