Fully Hilbertian Fields

TL;DR

The paper introduces fully Hilbertian fields, a stronger class than Hilbertian fields, which retain desirable properties and enable stronger Galois theoretic results, especially for uncountable fields.

Contribution

It defines fully Hilbertian fields, explores their properties, and demonstrates their advantages over Hilbertian fields in Galois theory, advancing the Jarden-Lubotzky twinning principle.

Findings

Fully Hilbertian fields exhibit similar behavior to Hilbertian fields.

They are more natural for uncountable fields.

They enable stronger Galois theoretic results.

Abstract

We introduce the notion of fully Hilbertian fields, a strictly stronger notion than that of Hilbertian fields. We show that this class of fields exhibits the same good behavior as Hilbertian fields, but for fields of uncountable cardinality, is more natural than the notion of Hilbertian fields. In particular, we show it can be used to achieve stronger Galois theoretic results. Our proofs also provide a step toward the so-called Jarden-Lubotzky twinning principle.