Partially ordered rings with negative squares

Abstract

A partially ordered ring (R, ≤) with negative squares is a partially ordered ring in which for some element a ∈ R, a ² < 0. In this paper, we study this class of partially ordered rings. It is shown that, for instance, for a totally ordered field K, the complex field C = K(i) over K, where i ² = −1, has a directed partial order with negative squares if and only if K is not algebraic over ℚ.