On power bases for rings of integers of relative Galois extensions

Let k be a finite extension of Q and L be an extension of k with rings of integers O-k and O-L, respectively. If O-L=O-k[theta], for some theta in O-L, then O-L is said to have a power basis over O-k. In this paper, we show that for a Galois extension L/k of degree p(m) with p prime, if each prime ideal of k above p is ramified in L and does not split in L/k and the intersection of the first ramification groups of all the prime ideals of L above p is non-trivial, and if p-1 inverted iota 2[k:Q], then O-L does not have a power basis over O-k. Here, k is either an extension with p unramified or a Galois extension of Q, so k is quite arbitrary. From this, for such a k the ring of integers of the nth layer of the cyclotomic Z(p)-extension of k does not have a power basis over O-k, if (p, [k:Q])=1. Our results generalize those by Payan and Horinouchi, who treated the case k a quadratic number field and L a cyclic extension of k of prime degree. When k=Q, we have a little stronger result.