A field $ K $ is pseudo-finite if it satisfies the following three conditions:

  • $ K $ is perfect
  • $ \operatorname{Gal}(K) $ is isomorphic to $ \hat{\mathbb{Z}} $ the inverse limit of $ \mathbb{Z}/n\mathbb{Z} $. Equivalently, $ K $ has a unique Galois extension of each degree.
  • $ K $ is pseudo-algebraically-closed (PAC), i.e., for every absolutely irreducible variety $ V \subset \mathbb{A}^n_K $, there is at least one $ K $-point on $ V $, i.e., $ V(K) \ne \emptyset $. Since $ K $ is perfect, we may replace "absolutely irreducible" (i.e., geometrically integral) with "geometrically irreducible." In fact, it suffices to check the case of absolutely irreducible plane curves.

Pseudo-finite fields form an elementary class. They were introduced by James Ax, who gave the following alternative characterization: pseudo-finite fields are exactly the infinite models of the theory of finite fields. Here, the "theory of finite fields" is the set of all first order statements which hold in all finite fields. Moreover, Ax showed that the theory of finite fields is recursively axiomatizable, by a variation of the above axioms which weakens each instance of the PAC condition (stated for plane curves), to allow for finitely many finite fields as counter-examples.

In particular, any non-principal ultraproduct of finite fields is pseudo-finite. The main difficulty in proving this fact lies in showing that such non-principal ultraproducts satisfy the PAC condition. This comes from Weil's Riemann Hypothesis for Curves.

Aside from ultraproducts of finite fields, other examples of pseudo-finite fields include:

  • Any infinite subfield of $ \mathbb{F}_p^{alg} $ having the correct Galois group, e.g. the subfield generated by $ \mathbb{F}_{p^2} $, $ \mathbb{F}_{p^3} $, $ \mathbb{F}_{p^5} $, and so on.
  • If $ \sigma $ is a Haar-random element of $ \operatorname{Gal}(K^{alg}/K) $ for $ K $ a global field or a finite field, then the fixed field of $ \sigma $ will be pseudo-finite, with probability 1.

For $ K $ pseudo-finite, let $ Abs(K) $ denote the subfield of $ K $ consisting of elements algebraic over the prime field. One has the following criterion for elementary equivalence of pseudo-finite fields:

Two pseudo-finite fields $ K $ and $ L $ are elementarily equivalent if and only if $ Abs(K) $ is isomorphic to $ Abs(L) $.
The possibilities for $ Abs(K) $ are essentially the fields of the form $ K^\sigma $, where $ \sigma $ is an automorphism of $ K $, and $ K $ is the algebraic closure of $ \mathbb{Q} $ or a finite field.

Ax's result characterizing pseudo-finite fields as the infinite models of the theory of finite fields has two difficult (deep) steps:

  • Proving that any non-principal ultraproduct of finite fields is pseudo-finite, specifically, showing that it is PAC. This uses Weil's Riemann Hypothesis for curves.
  • Proving that every pseudo-finite field is elementarily equivalent to an ultraproduct of finite fields (in fact, an ultraproduct of fields of prime order). Modulo the criterion for elementary equivalence of pseudo-finite fields, this ends up using the Chebotarev Density Theorem from Class Field Theory.

Pseudo-finite fields are closely connected to ACFA, the model companion of difference fields. If $ (K,\sigma) $ is a model of ACFA, then the fixed field of $ \sigma $ is always pseudo-finite. Up to elementary equivalence, all pseudo-finite fields arise in this way. Hrushovski’s work on the elementary theory of the Frobenius can be seen as generalizing Ax’s work from pseudo-finite fields to models of ACFA.

The theory of pseudo-finite fields is model complete after adding predicates $ SOL_n(x_1,\ldots,x_n) $, to be interpreted as $ \exists y : y^n = x_1 y^{n-1} + \cdots + x_n $. In the ring language, pseudo-finite fields are not model complete.

Pseudo-finite fields are always simple, but not stable. Forking can be described in some way similar to how it can be described in ACFA.

Pseudo-finite fields have strong definable Euler characteristics taking values in $ \mathbb{Z}/n\mathbb{Z} $ (definable with parameters), arising from the mod $ n $ counting functions for finite fields.