Totally positive units and squares

Abstract:

Let $K$ be a finite cyclic extension of the rational number field $Q$, with Galois group $G(K/Q)$ of order ${p^a}$ for an odd prime $p$. Armitage and Fröhlich [1] proved that if the order of 2 modulo $p$ is even and the class number ${h_K}$ of $K$ is odd then $U_K^ + = U_K^2$, where ${U_K}$ is the group of units of the ring of integers ${\mathcal {C}_K}$ of $K$, $U_K^ +$ is the group of totally positive units, and $U_K^2$ 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 ${Q.^2}$