Idele-Class Factor Sets and Class Field Theory

Abstract

requirements. His construction starts with the factor set obtained from the class field theory and yields the desired one in RK as the result of an analysis of the cohomology structure of TK, among others. In the present paper, we wish to propose a direct construction which leads to the same factor set (whence to the same extension group). For this purpose, we begin with a general analysis of idble-class factor sets in algebraic number fields and show that they behave just as element- or idele-factor sets. The starting point is that the 1-dimensional Galois cohomology groups in idele-classes vanish, which amounts more or less to a formulation of Noether's principal genus theorem [14] in terms of ideles and idble-classes. From auxiliary cyclic extensions, a certain canonical idele-class factor set can then be obtained, just as in the p-adic case (cf. [7]). It turns out that our canonical factor sets, thus obtained, coincide with Weil's. Although we do not claim, in the least, that all of Weil's results are obtained by our method, it seems to the writer that our construction is also of use. It also enables us to clarify the position of the canonical factor set among others. Another feature of our approach is that it does not preassume the reciprocity law in full. On the contrary, we may use our factor sets in proving the isomorphism and reciprocity theorems. More precisely, if we assume an index relation, the theory of cyclotomic fields, and the sum-relation of Hasse's invariants of the Brauer algebra classes, these theorems are derived from our factor sets and their construction just as in the local class field theory, by means of a homomorphism introduced in [10] (cf. also [1], [7], [11], [15]). Another application of our construction is a (partial) solution of a. problem concerning a 3-cohomology class which clarifies the arithmetical meaning of 3-cohomology. The writer is grateful to G. Hochschild for his friendly cooperation during the preparation of the present paper. A further study, aimed at idele-class cohomology in general, will be pursued in a joint paper by him and the writer.