Abstract :
Let p be a prime number. We say that a number field F satisfies the condition image when for any cyclic extension N/F of degree p, the ring image of p-integers of N has a normal integral basis over image. It is known that F=Q satisfies image for any p. It is also known that when pless-than-or-equals, slant19, any subfield F of Q(ζp) satisfies image. In this paper, we prove that when pgreater-or-equal, slanted23, an imaginary subfield F of Q(ζp) satisfies image if and only if image and p=43, 67 or 163 (under GRH). For a real subfield F of Q(ζp) with F≠Q, we give a corresponding but weaker assertion to the effect that it quite rarely satisfies image.
