On the Galois module structure over CM-fields
In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is that unramified realizations of a given finite abelian group Δ as a Galois group Gal (N/K) of an extension N of a given CM-field K are invariant under the involution on the set of all realizations of Δ over K which is induced by complex conjugation on K and by inversion on Δ. We give various implications of this result. For example, we show that the tame realizations of a finite abelian group Δ of odd order over a totally real number field K are completely characterized by ramification and Galois module structure.