Factor-sets of a group in its abstract unit group

Abstract

Presented to the Society, February 22, 1941; received by the editors August 24, 1940. (1) S. MacLane and 0. F. G. Schilling, Normal algebraic number fields, these Transactions, vol. 50 (1941), pp. 295-384. The present paper is entirely independent thereof. To indicate briefly what the connection is, let K be a real algebraic field normal of degree n over the rational field k, and let r be its Galois group. A theorem of Minkowski (Nachrichten der Gesellschaft der Wissenschaften zu G6ttingen, 1900, p. 90) asserts the existence of a unit H in K such that the group 4 generated by H and all its conjugates HO is a free abelian group of rank n -1, the only relation satisfied by the LU being that asserting that the norm of H is 1. This group 5 is of finite index in the group of all units of K. MacLane and Schilling study the class field theory of K by introducing crossed products (K, r, F). (K, r, F) is a normal simple algebra over k, determined by K, r, and a factor-set Fof elements F, TO, of K satisfying (0.3). Under the homomorphism F,,--(F,,7) carrying numerical factor-sets into factor-sets of principal ideals, those mapped into the identity are precisely the factor-sets of units. By further devices they are enabled in some cases to reduce the consideration of the latter to that of factorsets lying in the Minkowski subgroup t5.