Frank O. Wagner

Fields of Finite Morley Rank

If K is a field of finite Morley rank, then for any parameter set A ⊆ K eq the prime model over A is equal to the model-theoretic algebraic closure of A . A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl(∅).

Nilpotent complements and Carter subgroups in stable ℜ-groups

The following theorems are proved about the Frattini-free componentGΦ of a soluble stable ℜ-group: a) If it has a normal subgroupN with nilpotent quotientGΦ/N, then there is a nilpotent subgroupH ofGΦ withGΦ=NH. b) It has Carter subgroups; if the group is small, they are all conjugate. c) Nilpotency modulo a suitable Frattini-subgroup (to be defined) implies nilpotency. The last result makes use of a new structure theorem for the centre of the derivative of the Frattini-free component of a centreless soluble ℜ-group.

Die böse Farbe

We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariski-dimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a longstanding open question by Zilber.

Bad Fields in Positive Characteristic

If there are infinitely many p -Mersenne prime numbers, there is no bad field of positive characteristic p .

The free roots of the complete graph

There is a model-completion T n of the theory of a (reflexive) n-coloured graph such that R n is total, and R i o Rj ⊆ R i+j for all i,j. For n > 2, the theory T n is not simple, and does not have the strict order property. The theories T n combine to yield a non-simple theory Too without the strict order property, which does not eliminate hyperimaginaries.

Subgroups of Stable Groups

We define the notion of generic for an arbitrary subgroup H of a stable group, and show that H has a definable hull with the same generic properties. We then apply this to the theory of stable fields.

Model Theory with Applications to Algebra and Analysis: Counting and dimensions

We prove a theorem comparing a well-behaved dimension notion to a second, more rudimentary dimension. Specialising to a non-standard counting measure, this generalizes a theorem of Larsen and Pink on an asymptotic upper bound for the intersection of a variety with a general finite subgroup of an algebraic group. As a second application we apply this to bad fields of positive characteristic, to give an asymptotic estimate for the number of Fq-rational points of a definable multiplicative subgroup similar to the Lang-Weil estimate for curves over finite fields.

Applications of the group configuration theorem in simple theories

We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective geometry over a finite field.

Small Stable Groups and Generics

We define an 91-group to be a stable group with the property that a generic element (for any definable transitive group action) can only be algebraic over a generic. We then derive some corollaries for 91-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are 91-groups.

CONSTRUCTING AN ALMOST HYPERDEFINABLE GROUP

This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable (poly-)structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.