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We investigate some properties of (geometric) fields in toposes of sheaves over Boolean spaces and establish the internal validity of a number of classical theorems from Algebraic Geometry and the theory of ordered fields. We then use our results to obtain, via sheaf representations, some know theorems about (von Neumann) regular rings as well as some new theorems for regular f-rings. By contrast with previous investigations in these last two subjects (Saracino and Weispfenning {39} and van den Dries {42}) a more natural approach, inspired by work of Macintyre {30}, Loullis {29}, Bunge-Reyes {7} and Bunge {4},{5} is employed here. In addition to sheaf theoretic methods we use a variety of logical methods from geometric logic, infinitary intuitionistic logic and model theory. We also prove some new theorems on the transfer of subobjects along certain morphisms and a "lifting theorem" taking truth from statements about global sections to their internal validity.

No description available.

No description available.