Decidable Fragment of Theories in Field Arithmetic

TL;DR

This paper establishes the decidability of certain logical theories related to Hilbertian and PAC fields in characteristic 0, advancing the understanding of their logical properties.

Contribution

It proves the decidability of specific $orall^1 orall^1$ and $orall^1 exists^1$ theories for classes of Hilbertian and PAC fields in characteristic 0, which was previously unknown.

Findings

Decidability of $orall^1 orall^1$ theories for Hilbertian fields with characteristic 0.

Decidability of $orall^1 exists^1$ theories for Hilbertian and PAC fields in characteristic 0.

Extension of decidability results to perfect Hilbertian fields.

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.