Decidable Fragment of Theories in Field Arithmetic

In this paper, we show that the $\exists \forall$ theories of Hilbertian fields with charateristic 0 and perfect Hilbertian fields are both decidable. We also prove that the $\forall \exists$ theories of Hilbertian fields with charateristic 0, Hilbertian fields, PAC fields with characteristic 0, and PAC fields are all decidable.Comment: 15 page