This paper makes some insightful observations about geometric fields in the sense of E. Hrushovski and A. Pillay [Isr. J. Math. 85, No. 1–3, 203–262 (1994; Zbl 0804.03024)], and doing so, clarify certain questions on these fields. Remarkably, different concepts around these fields turn out to be equivalent.
From the abstract : “Let \(T\) be a complete theory of fields, possibly with extra structure. Suppose that model-theoretic algebraic closure agrees with field-theoretic algebraic closure, or more generally that model-theoretic algebraic closure has the exchange property. Then \(T\) has uniform finiteness, or equivalently, it eliminates the quantifier \(\exists^\infty\). It follows that very slim fields in the sense of Junker and Koenigsmann are the same things as geometric fields in the sense of Hrushovski and Pillay. Modulo some fine print, these two concepts are also equivalent to algebraically bounded fields in the sense of van den Dries. ”
The author also address some cardinality issues and obtain, notably, the following. The authors call a complete theory of fields “geometric” if it is as in the abstract quoted above, with model-theoretic algebraic closure having the exchange property.
Proposition 3.1. Fix a complete geometric theory \(T\) expanding the theory of fields, not necessarily algebraically bounded. If \(K\models T\) and \(X\subseteq K^n\) is an infinite definable set, then \(\vert X\vert=\vert K\vert\).
Proposition 3.6. Let \(T\) be a complete, geometric theory of infinite fields, possibly with extra structure.

(1)

If \(T\models T\) and \(X\) is an interpretable set of positive dimension, then \(\vert X\vert=\vert K\vert\).

(2)

If the language is countable, there is a model \(K\models T\) of cardinality \(\aleph_1\) such that every zero-dimensional interpretable set is countable.

Corollary 4.3, If \(T\) is a geometric theory of infinite fields in a countable language, then there is an uncountable model \(K\models T\) with countably many finite extensions of each degree \(n\) for each \(n\).