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 extend this result to $\kappa $ -prime models, for $\kappa $ an uncountable cardinal or $\aleph _\varepsilon $ .

Abstract We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group which is geometrically represented in a pseudo-finite field must be abelian.

Abstract Suppose p is the generic type of a differential-algebraic jet space to a finite dimensional differential-algebraic variety at a generic point. It is shown that p satisfies a certain strengthening of almost internality to the constants. This strengthening, which was originally called “being Moishezon to the constants” in [9] but is here renamed preserving internality to the constants, is a model-theoretic abstraction of the generic behaviour of jet spaces in complex-analytic geometry. An example is given showing that only a generic analogue holds in the differential-algebraic case: there is a finite dimensional differential-algebraic variety X with a subvariety Z that is internal to the constants, such that the restriction of the differential-algebraic tangent bundle of X to Z is not almost internal to the constants.

The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data: • The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field). • The degree of imperfection of K. • The first-order theory, in a suitable ω-sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K. They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.

Let G be an infinite abelian p-group and let K be a field of characteristic ≠ p. Let K[G] be the set of all sums Σg∈Gagg where the ag are in K, and all but finitely many ag are 0. Then K[G] is a K-algebra, with multiplication induced by the group multiplication on G. If G is countable, then the isomorphism type of K[G] has been completely described by S. D. Berman [1]. If G is a direct sum of countable groups, one can also describe K[G], as K[⊕iGi] ≃ ⊗iK[Gi]. If K contains all pn-th roots of unity, then K[G] is isomorphic to the ring of continuous functions from a Boolean space X to the field K with the discrete topology. In that case, the group UK[G] of invertible elements of K[G] is isomorphic to the direct sum of ∣G∣ copies of K×. More generally, if K is of the second kind with respect to p (see below for the definition), the group UK[G] has a simple description. Consider the subgroup SK[G] of elements Σgagg which have order a power of p and such that Σg ag = 1. This group is of course much simpler than UK[G]. Classifying SK[G] up to isomorphism reduces to the case where G has no element of infinite height, see [7]. If G is a direct sum of cyclic groups then the isomorphism type of SK[G] has been completely determined, in [2, 3, 7, 8]. The aim of this note is to show that a similar result is in general not possible for uncountable G. We use an invariant Γ associated to abelian groups, and for any regular uncountable cardinal κ, exhibit 2κ groups G for which Γ(G) = Γ(SK[G]) are pairwise distinct. Our work is based on a construction of Shelah [9], who constructed 2κ non-isomorphic abelian p-groups of cardinality κ for κ an uncountable regular cardinal.

In [1], examples of types of U-rank 1 (i.e., minimal types) in the theories of separably closed fields were constructed, en route to displaying certain dimension phenomena. We construct here additional examples with U-rank 1 and of various transcendence degrees over arbitrary separably closed fields. Our examples include ones which are minimal but of infinite transcendence degree, i.e., not thin. Our interest in building new examples was piqued after seeing the role played by minimal types over separably closed fields in Hrushovski's analysis of abelian varieties. This article is the result of several working sessions between the authors at Wesleyan University and Paris 7, and was completed during the Model Theory of Fields program at MSRI in 1998. We are grateful for the hospitality and support of all three institutions. We thank Elisabeth Bouscaren and Françoise Delon for reading an earlier version of this paper, providing useful suggestions and corrections.

Let K be a field of characteristic p. The map τ(X) = Xp − X is an additive endomorphism of K, with kernel Fp. The Galois extensions of K of order p are obtained by adjoining to K solutions to equations of the form Xp − X = a for some a in K. These extensions are called the Artin-Schreier extensions of K and have a cyclic Galois group. The study of Artin-Schreier extensions is very important for studying fields of characteristic p, in particular for studying valued fields of the form K((t)). An attempt at getting quantifier elimination for those fields would necessitate the adjunction to the language of fields of a cross-section for the function τ, i.e. a function σ such that τ ∘ σ is the identity on the image of τ. When K = Fp, such a cross-section is in fact definable in K((t)): it associates to τ(x) the element of {x, x + 1, …, x + p – 1} whose constant term is 0 (see [2]). When K is infinite, such a cross-section is usually not definable. The results presented in this paper originate from a question of L. van den Dries: is there a natural way of defining a cross-section σ for τ on F̃p, and is the theory of (F̃p, σ) decidable? (F̃p is the algebraic closure of Fp.)