Hilbert-Speiser number fields and Stickelberger ideals

Let p be a prime number. We say that a number field F satisfies the condition (H ′ pn) when any abelian extension N/F of exponent dividing p has a normal integral basis with respect to the ring of p-integers. We also say that F satisfies (H ′ p∞) when it satisfies (H ′ pn) for all n ≥ 1. It is known that the rationals Q satisfy (H ′ p∞) for all prime numbers p. In this paper, we give a simple condition for a number field F to satisfy (H ′ pn) in terms of the ideal class group of K = F (ζpn) and a “Stickelberger ideal” associated to the Galois group Gal(K/F ). As an application, we give a candidate of an imaginary quadratic field F which has a possibility of satisfying the very strong condition (H ′ p∞) for a small prime number p.