A note on the non-existence of prime models of theories of pseudo-finite fields

TL;DR

This paper proves that for non-pseudo-finite fields, no prime models exist for their pseudo-finite theories, and extends the result to larger cardinalities under GCH.

Contribution

It establishes the non-existence of prime models of pseudo-finite theories over non-pseudo-finite fields, generalizing to uncountable cardinals assuming GCH.

Findings

No prime models over non-pseudo-finite fields.

Generalization to ppa-prime models under GCH.

Results impact the understanding of model existence in field theories.

Abstract

We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we generalise this result to \kappa-prime models, for \kappa a regular uncountable cardinal or \aleph_\epsilon.