Totally positive units and squares

Abstract

Let K K be a finite cyclic extension of the rational number field Q Q , with Galois group G ( K / Q ) G(K/Q) of order p a {p^a} for an odd prime p p . Armitage and Fröhlich [1] proved that if the order of 2 modulo p p is even and the class number h K {h_K} of K K is odd then U K + = U K 2 U_K^ + = U_K^2 , where U K {U_K} is the group of units of the ring of integers C K {\mathcal {C}_K} of K K , U K + U_K^ + is the group of totally positive units, and U K 2 U_K^2 is the group of unit squares. The purpose of this paper is to provide a generalization of this result to a larger class of abelian extensions of Q . 2 {Q.^2}