On formally real Division Algebras and Quasifields of Rank two

In view of Albert's classical result that the dimension of each ordered, associative proper algebra over its center is necessarily infinite (cf. [1]), it seems not unlikely that a similar statement also holds for the rank of each proper formally real quasifield F, i.e. for the dimension of F over its kernel. Indeed, for some classes of ordered near fields and for ordered quasifields admitting a real kernel, D. Gröger [3] and J. Joussen [4] were able to verify that their rank must be one or infinite. However, making use of valuation theoretical means, recently (cf. [7]) we were able to show that for each natural number n formally real quasifields of rank n do exist. In this note we will present a more elementary approach to a special case of the problem in question, namely to the existence of formally real quasifields and of formally real (unitary, necessarily not associative) division algebras of rank two.