Introduction. Let N/K be a finite Galois extension of number fields of group G. Let Ko(Z[G]) denote the Grothendieck group of finitely generated projective (left) 7[G]-modules, with Cl(Z[G]) the subgroup of elements which have rank 0. In [C1] Chinburg used the four term exact sequences of prescribed extension class introduced by Tate in ([T2], pp. 56-57) to define an invariant Qm(N/K) E Cl(Z[G]) of the multiplicative Galois structure of N. (The class Qm(N/K) is also often denoted Q(N/K, 3).) This invariant can be described in terms of local and global Weil groups, and has been conjectured by Chinburg to equal the Cassou-Nogues-Frohlich root number class [C2]. There is by now a considerable amount of evidence to support this conjecture (cf. [D] ?4). The conjecture is in part motivated by Tate's re-formulation of the Stark conjecture and, specifically, his construction of a natural generalization of the classical Dirichlet unit regulator [T2]. More recently, Frohlich has defined an integral version of the Stark-Tate regulator, and by integrally reinterpreting much of the Tate approach has studied elements in natural quotients of Cl(Z[G]) which explicitly relate the Galois structure of unit lattices and (the regular part of) ideal class groups (cf. [F3], [F4], ([D], ?3)). In this paper we shall show that the results of Fr6hlich represent approximations to the conjecture of Chinburg. For example, we show that the 'fine structure' results of [F4] are for the special classes of abelian extensions considered there the best approximations to a proof of Chinburg's conjecture which can be obtained without analysis of extension class data. Moreover, for certain classes of field extensions the relevant cohomological questions can be resolved and so we shall obtain a full proof of the Chinburg conjecture for these extensions. In this way the dependence of the conjecture on the role of the canonical cohomology classes of Tate in defining Qm(N/K) is made very clear. Also, taken in conjunction with recent work of Snaith [Sn] on the additive Chinburg invariant Q(N/K, 2), a special case of our results provides an infinite family of (wildly ramified) abelian extensions for which all of the conjectures of Chinburg are now completely verified (cf. Remark 1.11(ii)).