Non-archimedean directed fields K(i) with o-subfield K and i2=−1

Let K be a linearly ordered field, and let i be a root of the equation x2+1=0. If K is archimedean, it is known that K(i) cannot be a 2 dimensional directed algebra over K. For non-archimedean K, however, Yang (2006) [17] proved the existence of directed fields K(i) that are 2 dimensional directed algebras over K. In this paper, we characterize directed fields of the form K(i) that extend the order of K.