Introduction. In Hilbert’s Zahlbericht one can find the first global result on the Galois module structure of rings of integers ([H], Satz 132). Slightly extended it states that if N/Q is a tame extension with abelian Galois group ∆, then oN , the ring of integers in N , is free as a module over the group ring Z∆; moreover, an explicit canonical algebraic integer a such that oN = Z∆a can be given. The numbers a (δ ∈ ∆), the algebraic conjugates of a, are said to form a normal integral basis of the field extension N/Q. This result has been the starting point for a modern development, which has led to the deep result that the structure of oN as a module over Z∆, for arbitrary tame Galois extensions of number fields N/K with Galois group ∆, is determined in terms of the symplectic root numbers of N/K (see [F] and [T1]). Frohlich’s book [F] also contains a detailed introduction and a rather complete list of references. For a more recent survey on Galois module theory we refer to [Ca–Ch–F–T]. In the special case that ∆ is of odd order the result mentioned above gives that oN is a free Z∆-module, but the proof does not provide an explicit basis. If one considers the richer oK∆-module structure of oN rather than the Z∆-module structure alone, then oN is expected to be “usually” not even free if K 6= Q. Results of Taylor show that by modifying both oN and oK∆ one can sometimes—if K and N are certain ray class fields over imaginary quadratic number fields—achieve the “ideal” of free modules with explicit generators (see [C–T]). However, if one decides not to modify the original classical problem of the determination of the oK∆-module structure of oN , then a natural question is to what extent, for given K and ∆, the realization of ∆ as a Galois group of a tame extension N/K is determined by the ramification of N/K together with the structure of oN as an oK∆-module. This point of view is worked out in [B2]. In a sense the core of the question is how rare unramified extensions which possess a normal integral basis are. We mention in passing that this question is equivalent to a special case of a problem considered by Taylor in recent work, that of determining the kernel