Let T be the theory of fields K of characteristic 0 such that the equation x^4 + y^4 = 1 has only four solutions in K. We show that T has a model companion. More generally, if K_0 is a field of characteristic 0 and C is a curve (affine or projective) of genus ≥ 2 with C(K_0) = ∅, then there is a model companion CXF of the theory of fields K extending K_0 with C(K) = ∅. We can use this theory to construct a field K with […]
Joint work with Samaria MontenegroThe striking resemblance between the behaviour of pseudo-algebraically closed, pseudo real closed and pseudo p-adically fields has lead to numerous attempts at describing their properties in a unified manner. In this talk I will present another of these attempts: the class of pseudo-T-closed fields, where T is an enriched theory of fields. These fields verify a “local-global” principle with respect to models of T for the existence of points on varieties. Although it very much resembles previous such attempts, our approach is more model theoretic in […]
