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The paper deals with the following problems concerning pseudo-ordered denumerable fields:    (i)  a connection between Fermat’s two squares theorem and the unique pseudo-order in a finite field;    (ii)  properties of a proper pseudo-order determined by any prime number in the field of rational numbers;    (iii)  existence of a proper pseudo-order in every subfield of the sequence used to obtain the field of constructible numbers;    (iv)  some brief of applications of the latter pseudo-orders to construct new algebraic and geometric structures.    In particular, we extend the known construction of finite nearfields or quasifields given by e.g. W. A. Pierce or P. Dembowski – to infinite cases.

Some examples of denumerable pseudo-ordered fields and their applications to geometry