Decidable Fragment of Theories in Field Arithmetic

Abstract

In this paper, we show that the ${\exists }^1{\forall }^1$ theories of Hilbertian fields with charateristic 0 and perfect Hilbertian fields are both decidable. We also prove that the ${\forall }^1{\exists }^1$ theories of Hilbertian fields with charateristic 0, Hilbertian fields, PAC fields with characteristic 0, and PAC fields are all decidable.

15 pages