On algebraic closure in pseudofinite fields

We study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

Random hypergraphs in pseudofinite fields

We prove thatn-hypergraphs can be interpreted in e-free perfect PAC fields in particular in pseudofinite fields. We use methods of function field arithmetic, more precisely we construct generic polynomials with alternating groups as Galois groups over a function field.

Coding complete theories in Galois groups

In this paper, I will give a new characterisation of the spaces of complete theories of pseudofinite fields and of algebraically closed fields with a generic automorphism (ACFA) in terms of the Vietoris topology on absolute Galois groups of prime fields.

From Stability to Simplicity

§1. Introduction. In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular algebraic structures which have been studied recently (pseudofinite fields, algebraically closed fields with a generic automorphism, smoothly approximable structures) turn out to be simple. The interest is basically that a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. Stable theories will be defined formally in the next section. An exhaustive study of them is carried out in [33]. Without trying to read Shelah's mind, we feel comfortable in saying that the importance of stability for Shelah lay partly in the fact that an unstable theory T has 2λ many models in any cardinal λ ≥ ω1 + |T| (proved by Shelah). (Note that for λ ≥ |T| 2λ is the maximum possible number of models of cardinality λ.)