Normal Algebraic Number Fields.

Introduction. In this paper we present a detailed account of the results recently published in the Proceedings of the National Academy of Sciences [29 Our theory is an attempt to generalize the results of the classical class field theory to arbitrary normal fields. In the last analysis, the theory of cyclic extensions Z of an algebraic number field k can be described in terms of cyclic algebras (Z/k, X, a) and collections of local algebras (Zp/kp, 5, ap), for all prime divisors p of the base field. As a matter of fact, Chevalley's new approach [ll ] to the classical theory by means of ideal elements may be viewed in the light of our assertion (§8). It is a well-known fact that for arbitrary normal fields K/k with the Galois group T, the crossed products (K/k, T, F), where F denotes a factor set for T in K, are the strict analogues of the cyclic algebras. We use this feature of normal fields and their associated algebras to extend the classical (cyclic) theory. In this generalization ideal elements are replaced by "ideal algebras," where an ideal algebra is a collection of local algebras, one for each p-adic extension Kp/kp (§4). It might be conjectured from this approach that most of the results of the classical theory may easily be generalized, but this is not the case(2). Entirely new problems, mostly group-theoretical ones, block the path which has been envisaged by various statements of E. Noether [32, 33] on noncommutative methods. One of our final theorems may suffice to illustrate the rather unexpected results of this paper. Our new "class group" of ideal algebras can be represented as F%'/\F")TW, where F2T denotes the group of factor sets of ideals relatively prime to the different of K/k, where TW is the group of transformation sets of such ideals, and where (F") are principal ideals generated by "norm residues" (§26). For an abelian (non-cyclic) field this class group is not isomorphic to the Galois group, as in the ordinary theory; it is rather a cyclic group whose order is equal to the least common multiple of the orders of the elements in the Galois group. In other words, for arbitrary abelian fields our theory does not give the classical law of reciprocity. Only a composition which is extraneous to the theory of factor sets yields the usual theory for general abelian extensions. In the first part of this paper we follow an unpublished investigation of E. Artin (§5). It deals with the theory of p-primary factor sets, that is, factor sets whose elements involve only the divisors of a fixed prime divisor p of k.