UNRAMIFIED ABELIAN EXTENSIONS OF CM-FIELDS AND THEIR GALOIS MODULE STRUCTURE JAN BRINKHUIS Introduction In a previous paper [1], a non-existence result was given for normal integral bases of abelian extensions of CM-fields N/K which are unramified at all finite primes. The aim of this paper is to give an account which gives, moreover, some information on the class (o^),, A of the o K A-module o^ in the locally free class group Cf(o K A) and on its order, where A = Gal (N/K). To this end we define a certain subgroup of C7(o K A) which contains (o^),, A , a homomorphism from this subgroup to a group with a transparent structure, and then we compute the image of (o^),, A . The outcome generalizes the result in [1]. Surprisingly, in the present approach one does not need the assumption that the top field N is a CM-field. Therefore one obtains a much stronger result than in [1]: see Theorem 2.9 and Corollary 2.10. 1. A lifting of the Galois module class For each number field F, let o F (respectively V F , respectively A F ) be the ring of integers (respectively of adeles, respectively of integral adeles of F; we shall not include the infinite primes). By the Mayer-Vietoris sequence of algebraic AT-theory associated to the following square of group rings of A over o F , F, A F and V F , (1.1) and by standard stability results of algebraic AT-theory, one obtains the following exact sequence which gives a description of the projective class group, or, what is here the same, the locally free class group of o F A 1 >o F A* >FA*®A F A* >V F A* >C7(o F A) >1, (1.2) where the first map is defined by x -*• (x, x) and the second by (y, z) -> y~ x z. See the account in Swan [4] for details. Received 19 September 1990. 1991 Mathematics Subject Classification 11R33 Bull. London Math. Soc. 24 (1992) 236-242