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 [ θ ], for some θ 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 ∤ 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 n th 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.