MODEL THEORY OF FINITE AND PSEUDO-FINITE FIELDS

Intensive course

The course is part of the activities of the graduate school MALJA.

Lecturer

Zoe Chatzidakis (Université Paris 7)

Lectures

Lectures take place in University of Helsinki, the Exactum building:

Wed, 22 Apr in C124 at 12-14
 Thu, 23 Apr in B322 at 12-14
 Fri, 24 Apr in B322 at 13-15

Mon, 27 Apr in B322 at 12-14
 Tue, 28 Apr in B322 at 12-14
 Wed, 29 Apr in C124 at 12-14

Course description

This course will study the elementary theory of finite fields, that is,
the set of sentences which are true in all finite fields. This theory
was described by Ax, and its infinite models are called pseudo-finite
fields. It is a decidable theory. The behaviour of pseudo-finite fields
gives you asymptotic information on the behaviour of finite fields, and
for this reason they are interesting. They are also easier to study.

We will first start by studying finite fields, and state the classical
theorems of Lang-Weil and of Tchebotarev. They are the main tools in the
axiomatisation of pseudo-finite fields. We will also describe the
completions of the theory of pseudo-finite fields. The algebraic notions
needed in the proof will be done fairly thoroughly. There will also be a
set of notes which will give additional details.

If time permits, we will discuss such concepts as algebraic closure,
definable sets, independence theorem, ranks, and some applications to
algebraic groups by Hrushovski and Pillay.

Contact

For financial support (for doctoral students of MALJA), contact
 <Lauri.Hella AT uta.fi>. For other practical questions, contact
 <Kerkko.Luosto AT Helsinki.FI>.