Affine super-Pythagorean geometry (Q1972381)

Super-Pythagorean fields were introduced by R. Elman, T. Y. Lam and L. Bröcker in 1972 within the framework of the theory of reduced Witt rings and spaces of orderings of fields. They can be defined as fields in which every half-ordering not containing \(-1\) is an ordering. In [Beitr. Algebra Geom. 39, 255-258 (1998; Zbl 0915.51012)] the author has studied the geometrical consequences of this notion, in the main by presenting an axiomatic system for Euclidean planes coordinatized by super-Pythagorean fields. In the note under review, the author improves and simplifies this axiomatic system by providing a geometrical (even an affine) pendant for the following algebraic characterization of super-Pythagorean fields \(F:-1\notin F^2\) and \(\emptyset\neq F^2\cap\{-x,1 +x,x(1+x)\}\) for all \(x\in F\).